diff --git a/_forest b/_forest
index 4902773..7650104 160000
--- a/_forest
+++ b/_forest
@@ -1 +1 @@
-Subproject commit 4902773fc1eaabce6cbe3a8d624379b5a4f8a077
+Subproject commit 76501041f98204820816ff40ff52bb6e11f3ac3d
diff --git a/_site/2023/04/09/gsi-applications.html b/_site/2023/04/09/gsi-applications.html
index 16d8540..2353b80 100644
--- a/_site/2023/04/09/gsi-applications.html
+++ b/_site/2023/04/09/gsi-applications.html
@@ -83,8 +83,8 @@ <h2 id="engr-101-f23">ENGR 101 (F23)</h2>
 
 <iframe width="560" height="315" src="https://www.youtube.com/embed/Cd9Tq7s_BhQ" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen=""></iframe>
 
-<style> .pdf-embed-wrap-a5025c74-6d89-461b-a199-2689d0b46488 { display: flex; flex-direction: column; width: 100%; height: 650px; } .pdf-embed-container-a5025c74-6d89-461b-a199-2689d0b46488 { height: 100%; } .pdf-link-a5025c74-6d89-461b-a199-2689d0b46488 { background-color: white; text-align: center; border-style: solid; } .pdf-embed-container-a5025c74-6d89-461b-a199-2689d0b46488 iframe { width: 100%; height: 100%; } </style>
-<div class="pdf-embed-wrap-a5025c74-6d89-461b-a199-2689d0b46488"> <div class="pdf-link-a5025c74-6d89-461b-a199-2689d0b46488"> <a href="/files/101-gsi-application.pdf" target="_blank">View PDF</a> </div> <div class="pdf-embed-container-a5025c74-6d89-461b-a199-2689d0b46488"> <iframe src="/files/101-gsi-application.pdf" frameborder="0" allowfullscreen=""></iframe> </div> </div>
+<style> .pdf-embed-wrap-c6b07f7a-3cd5-4207-9550-83fd90f7ac56 { display: flex; flex-direction: column; width: 100%; height: 650px; } .pdf-embed-container-c6b07f7a-3cd5-4207-9550-83fd90f7ac56 { height: 100%; } .pdf-link-c6b07f7a-3cd5-4207-9550-83fd90f7ac56 { background-color: white; text-align: center; border-style: solid; } .pdf-embed-container-c6b07f7a-3cd5-4207-9550-83fd90f7ac56 iframe { width: 100%; height: 100%; } </style>
+<div class="pdf-embed-wrap-c6b07f7a-3cd5-4207-9550-83fd90f7ac56"> <div class="pdf-link-c6b07f7a-3cd5-4207-9550-83fd90f7ac56"> <a href="/files/101-gsi-application.pdf" target="_blank">View PDF</a> </div> <div class="pdf-embed-container-c6b07f7a-3cd5-4207-9550-83fd90f7ac56"> <iframe src="/files/101-gsi-application.pdf" frameborder="0" allowfullscreen=""></iframe> </div> </div>
 <h2 id="eecs-376-f23">EECS 376 (F23)</h2>
 <p><strong>outcome</strong>: rejected. No response.</p>
 <h3 id="too-long-version">too long version</h3>
@@ -153,7 +153,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/04/10/miserliness.html b/_site/2023/04/10/miserliness.html
index b22b057..978d7ce 100644
--- a/_site/2023/04/10/miserliness.html
+++ b/_site/2023/04/10/miserliness.html
@@ -142,7 +142,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/04/14/rasterize-pdfs.html b/_site/2023/04/14/rasterize-pdfs.html
index 45f5808..8f70ab2 100644
--- a/_site/2023/04/14/rasterize-pdfs.html
+++ b/_site/2023/04/14/rasterize-pdfs.html
@@ -135,7 +135,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/04/23/scripting.html b/_site/2023/04/23/scripting.html
index 7ebef88..72ff7df 100644
--- a/_site/2023/04/23/scripting.html
+++ b/_site/2023/04/23/scripting.html
@@ -266,7 +266,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/05/02/chili-oil.html b/_site/2023/05/02/chili-oil.html
index 70ed2a5..70a4b6a 100644
--- a/_site/2023/05/02/chili-oil.html
+++ b/_site/2023/05/02/chili-oil.html
@@ -122,7 +122,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/05/07/nonstandard-analysis.html b/_site/2023/05/07/nonstandard-analysis.html
index e79fa85..28dcda1 100644
--- a/_site/2023/05/07/nonstandard-analysis.html
+++ b/_site/2023/05/07/nonstandard-analysis.html
@@ -116,7 +116,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/05/10/linux-graphics-modules.html b/_site/2023/05/10/linux-graphics-modules.html
index 6df018b..4b9dbbf 100644
--- a/_site/2023/05/10/linux-graphics-modules.html
+++ b/_site/2023/05/10/linux-graphics-modules.html
@@ -137,7 +137,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/07/24/book-catalogue.html b/_site/2023/07/24/book-catalogue.html
index 1773c2a..a368b73 100644
--- a/_site/2023/07/24/book-catalogue.html
+++ b/_site/2023/07/24/book-catalogue.html
@@ -291,7 +291,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/08/03/nic-bonding.html b/_site/2023/08/03/nic-bonding.html
index 6357260..77624f1 100644
--- a/_site/2023/08/03/nic-bonding.html
+++ b/_site/2023/08/03/nic-bonding.html
@@ -126,7 +126,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/10/10/tortoise-tts.html b/_site/2023/10/10/tortoise-tts.html
index b0f4818..057c3ea 100644
--- a/_site/2023/10/10/tortoise-tts.html
+++ b/_site/2023/10/10/tortoise-tts.html
@@ -252,7 +252,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/10/29/coordinating-processes.html b/_site/2023/10/29/coordinating-processes.html
index 6275ff7..8f7e39b 100644
--- a/_site/2023/10/29/coordinating-processes.html
+++ b/_site/2023/10/29/coordinating-processes.html
@@ -108,7 +108,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/11/07/ble-sh.html b/_site/2023/11/07/ble-sh.html
index 1b301a1..dffa191 100644
--- a/_site/2023/11/07/ble-sh.html
+++ b/_site/2023/11/07/ble-sh.html
@@ -106,7 +106,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/12/19/icecc.html b/_site/2023/12/19/icecc.html
index da1b1a1..f9bd117 100644
--- a/_site/2023/12/19/icecc.html
+++ b/_site/2023/12/19/icecc.html
@@ -131,7 +131,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2023/12/20/xmonad.html b/_site/2023/12/20/xmonad.html
index c9eef2b..d8b3112 100644
--- a/_site/2023/12/20/xmonad.html
+++ b/_site/2023/12/20/xmonad.html
@@ -193,7 +193,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/01/08/thunderbird-and-nix.html b/_site/2024/01/08/thunderbird-and-nix.html
index cd04993..7e59bee 100644
--- a/_site/2024/01/08/thunderbird-and-nix.html
+++ b/_site/2024/01/08/thunderbird-and-nix.html
@@ -170,7 +170,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/01/16/mellanox.html b/_site/2024/01/16/mellanox.html
index c1f621d..07b5032 100644
--- a/_site/2024/01/16/mellanox.html
+++ b/_site/2024/01/16/mellanox.html
@@ -110,7 +110,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/02/08/natural-models.html b/_site/2024/02/08/natural-models.html
index b6baeda..3bfaf5c 100644
--- a/_site/2024/02/08/natural-models.html
+++ b/_site/2024/02/08/natural-models.html
@@ -166,7 +166,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/02/12/noctua.html b/_site/2024/02/12/noctua.html
index fd04bfa..8515bb4 100644
--- a/_site/2024/02/12/noctua.html
+++ b/_site/2024/02/12/noctua.html
@@ -115,7 +115,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/02/24/vim.html b/_site/2024/02/24/vim.html
index 2f80b70..3e21216 100644
--- a/_site/2024/02/24/vim.html
+++ b/_site/2024/02/24/vim.html
@@ -123,7 +123,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/03/10/deoplete.html b/_site/2024/03/10/deoplete.html
index c56552e..d013ffa 100644
--- a/_site/2024/03/10/deoplete.html
+++ b/_site/2024/03/10/deoplete.html
@@ -166,7 +166,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/04/11/fountain-pen-nibs.html b/_site/2024/04/11/fountain-pen-nibs.html
index 457ded3..7c395a8 100644
--- a/_site/2024/04/11/fountain-pen-nibs.html
+++ b/_site/2024/04/11/fountain-pen-nibs.html
@@ -157,7 +157,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/2024/06/17/archiving-stack-overflow.html b/_site/2024/06/17/archiving-stack-overflow.html
index 7bfb998..7ae174a 100644
--- a/_site/2024/06/17/archiving-stack-overflow.html
+++ b/_site/2024/06/17/archiving-stack-overflow.html
@@ -155,7 +155,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/404.html b/_site/404.html
index 9528eca..82dcd92 100644
--- a/_site/404.html
+++ b/_site/404.html
@@ -100,7 +100,7 @@ <h1>404</h1>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/category-theory/2023/05/18/equivalence-of-categories.html b/_site/category-theory/2023/05/18/equivalence-of-categories.html
index c85ef59..dd84d69 100644
--- a/_site/category-theory/2023/05/18/equivalence-of-categories.html
+++ b/_site/category-theory/2023/05/18/equivalence-of-categories.html
@@ -153,7 +153,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/category/category-theory/index.html b/_site/category/category-theory/index.html
index f9ae0e0..2c005b2 100644
--- a/_site/category/category-theory/index.html
+++ b/_site/category/category-theory/index.html
@@ -90,7 +90,7 @@ <h1>Archive of posts with category 'category-theory'</h1>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/category/site-construction/index.html b/_site/category/site-construction/index.html
index 16db87c..e3ebf87 100644
--- a/_site/category/site-construction/index.html
+++ b/_site/category/site-construction/index.html
@@ -120,7 +120,7 @@ <h1>Archive of posts with category 'site-construction'</h1>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/feed.xml b/_site/feed.xml
index 760c187..df5c7ce 100644
--- a/_site/feed.xml
+++ b/_site/feed.xml
@@ -1,4 +1,4 @@
-<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" ><generator uri="https://jekyllrb.com/" version="4.3.3">Jekyll</generator><link href="https://hejohns.github.io/feed.xml" rel="self" type="application/atom+xml" /><link href="https://hejohns.github.io/" rel="alternate" type="text/html" /><updated>2024-06-29T04:12:55-04:00</updated><id>https://hejohns.github.io/feed.xml</id><title type="html">hejohns’ jekyll site</title><subtitle>a pure waste of space</subtitle><author><name>Johnson He</name><email>hejohns α umich δ edu</email></author><entry><title type="html">archiving stack overflow (for the forest)</title><link href="https://hejohns.github.io/2024/06/17/archiving-stack-overflow.html" rel="alternate" type="text/html" title="archiving stack overflow (for the forest)" /><published>2024-06-17T00:00:00-04:00</published><updated>2024-06-29T04:03:08-04:00</updated><id>https://hejohns.github.io/2024/06/17/archiving-stack-overflow</id><content type="html" xml:base="https://hejohns.github.io/2024/06/17/archiving-stack-overflow.html"><![CDATA[<h2 id="first-iteration">First Iteration</h2>
+<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" ><generator uri="https://jekyllrb.com/" version="4.3.3">Jekyll</generator><link href="https://hejohns.github.io/feed.xml" rel="self" type="application/atom+xml" /><link href="https://hejohns.github.io/" rel="alternate" type="text/html" /><updated>2024-07-02T03:34:02-04:00</updated><id>https://hejohns.github.io/feed.xml</id><title type="html">hejohns’ jekyll site</title><subtitle>a pure waste of space</subtitle><author><name>Johnson He</name><email>hejohns α umich δ edu</email></author><entry><title type="html">archiving stack overflow (for the forest)</title><link href="https://hejohns.github.io/2024/06/17/archiving-stack-overflow.html" rel="alternate" type="text/html" title="archiving stack overflow (for the forest)" /><published>2024-06-17T00:00:00-04:00</published><updated>2024-06-29T04:03:08-04:00</updated><id>https://hejohns.github.io/2024/06/17/archiving-stack-overflow</id><content type="html" xml:base="https://hejohns.github.io/2024/06/17/archiving-stack-overflow.html"><![CDATA[<h2 id="first-iteration">First Iteration</h2>
 
 <p>(Note: I call any stack exchange site “stack overflow”.)</p>
 
diff --git a/_site/forest/awodey-2010.xml b/_site/forest/awodey-2010.xml
index aeec504..711037f 100644
--- a/_site/forest/awodey-2010.xml
+++ b/_site/forest/awodey-2010.xml
@@ -1,10 +1,395 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>479</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>641</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>642</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+type="local"
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
+        classifier and a natural numbers object."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+type="local"
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
+type="local"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course, if we ignore natural numbers objects for now.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/blass-1988.xml b/_site/forest/blass-1988.xml
index 1ab8fc9..28d6bd8 100644
--- a/_site/forest/blass-1988.xml
+++ b/_site/forest/blass-1988.xml
@@ -1,66 +1,76 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>734</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
         computation and the geometric side of topos theory. We begin with a brief
         outline of the history and basic concepts of topos theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
         the history of topos theory, the relation between Grothendieck and
         Elementary topoi, and the relation between geometric and logical
         morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="blass-1988.pdf">blass-1988.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>508</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>735</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -81,27 +91,30 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>737</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -115,9 +128,10 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/blass-2003.xml b/_site/forest/blass-2003.xml
index eea3b9a..47bdf0c 100644
--- a/_site/forest/blass-2003.xml
+++ b/_site/forest/blass-2003.xml
@@ -1,29 +1,29 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>498</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>730</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
         instances are simply Herbrand valid. The characterization turns out to
         coincide with a known characterization of game semantical validity for
         multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     This is currently a "would like to read, but no urgency" paper, because
     Blass + Herbrand's theorem.
     <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+type="external"
+href="blass-2003.pdf">blass-2003.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/carette-kiselyov-shan-2007.xml b/_site/forest/carette-kiselyov-shan-2007.xml
index 542423f..00050b6 100644
--- a/_site/forest/carette-kiselyov-shan-2007.xml
+++ b/_site/forest/carette-kiselyov-shan-2007.xml
@@ -1,20 +1,20 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>569</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>720</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -35,14 +35,14 @@ root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Abstract</fr:ta
         stuck.  To achieve self-interpretation and show Jones-optimality, we
         relate this exemplar of higher-rank and higher-kind polymorphism to
         plugging a term into a context of let-polymorphic bindings.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -64,21 +64,28 @@ root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Notes</fr:taxon
         that it is the free syntax generated by the encoded rules/initial
         algebra/inductive dataype.
         But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>568</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>721</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -92,14 +99,14 @@ root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:ta
         computability</fr:strong>, logical relations are manipulated as if they were sets,
         making the essence of many complex logical relations arguments
         accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -112,15 +119,16 @@ root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</f
         and I just don't have some context, or it's actually presented almost
         mystically.
         The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+href="carette-kiselyov-shan-2007.xml"
+addr="carette-kiselyov-shan-2007"
+title="Finally Tagless, Partially Evaluated">finally tagless encodings</fr:link>
         and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
+type="external"
+href="hoffman-1999">a justification for hoas</fr:link> doesn't seem super
         on-the-point, but if it's mentioned it must be for good reason…
     </fr:p><fr:p>
         Again, quotes are air-quotes, not actual quotes.
         <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+type="external"
+href="sterling-2022.pdf">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/dummett-91.xml b/_site/forest/dummett-91.xml
index 78486c0..889a9b4 100644
--- a/_site/forest/dummett-91.xml
+++ b/_site/forest/dummett-91.xml
@@ -1,32 +1,34 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>582</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>770</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
+type="external"
+href="zeilberger">zeilberger</fr:link> quotes on his work on
         polarity.
     </fr:p><fr:p>
         What's the relationship between harmony-stability and Pfenning's local
         soundness and completeness?
         They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/ericgio.xml b/_site/forest/ericgio.xml
index 52d34fe..c2d90bc 100644
--- a/_site/forest/ericgio.xml
+++ b/_site/forest/ericgio.xml
@@ -1,247 +1,383 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>ericgio</fr:addr><fr:route>ericgio.xml</fr:route><fr:title>Eric Giovannini</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://public.websites.umich.edu/~ericgio/index.html"
-type="external">https://public.websites.umich.edu/~ericgio/index.html</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>ericgio</fr:addr><fr:route>ericgio.xml</fr:route><fr:title>Eric Giovannini</fr:title><fr:taxon>Person</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://public.websites.umich.edu/~ericgio/index.html">https://public.websites.umich.edu/~ericgio/index.html</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -249,7 +385,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -257,131 +394,211 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/forest.json b/_site/forest/forest.json
index 31d3098..4f26478 100644
--- a/_site/forest/forest.json
+++ b/_site/forest/forest.json
@@ -1 +1 @@
-{"ericgio":{"title":"Eric Giovannini","taxon":"Person","tags":[],"route":"ericgio.xml","metas":{}},"hejohns-000R":{"title":"BLANK","taxon":null,"tags":[],"route":"hejohns-000R.xml","metas":{}},"hejohns-000Q":{"title":"Every (elementary) topos is regular","taxon":"Lemma","tags":["scratch"],"route":"hejohns-000Q.xml","metas":{}},"hejohns-000P":{"title":"Logical Calculations","taxon":null,"tags":["scratch"],"route":"hejohns-000P.xml","metas":{}},"shoenfield-1967":{"title":"Mathematical Logic","taxon":"Reference","tags":[],"route":"shoenfield-1967.xml","metas":{}},"logicmatters":{"title":"Peter Smith's Book Reviews","taxon":null,"tags":[],"route":"logicmatters.xml","metas":{}},"hejohns-000O":{"title":"Basic information","taxon":null,"tags":[],"route":"hejohns-000O.xml","metas":{}},"hinman-2005":{"title":"Fundamentals of Mathematical Logic","taxon":"Reference","tags":[],"route":"hinman-2005.xml","metas":{}},"lambek-scott-1986":{"title":"Introduction to higher order categorical 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diff --git a/_site/forest/friedman-1975.xml b/_site/forest/friedman-1975.xml
index cd5f5dc..5b8c5ff 100644
--- a/_site/forest/friedman-1975.xml
+++ b/_site/forest/friedman-1975.xml
@@ -1,20 +1,20 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
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 show-heading="true"
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 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>578</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>712</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
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 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -31,57 +31,68 @@ root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:ta
         Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
         with or without types, have been considered. The motivation for introducing
         these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex
+display="inline"><![CDATA[⊢ s = t]]></fr:tex> defined
         in the text, is equivalent to one of these definitions of convertibility.
         It is easy to see that any two convertible terms define the same functional
         of finite type. We show here that any two non- convertible terms define
         different functionals of finite type.[This result was obtained in 1970.]
         This, coupled with the known decidability of convertibility, tells us that
         "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     Square brackets denote author's footnotes.
     <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="friedman-1975.pdf">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
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-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>576</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>713</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
+type="local"
 href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">Equality Between Functionals</fr:link> and <fr:link
 type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
 href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+addr="statman-1985"
+title="Equality Between Functionals Revisited">Equality Between Functionals Revisited</fr:link>.
     Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://proofassistants.stackexchange.com/a/1177">stack overflow</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>577</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>714</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
-href="friedman-1975.xml"
 type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">3</fr:link>) in the context of what we know today about the model
         theory of the typed λ-calculus. In order to do this, it is appropriate to
         survey recent work in this area, in so far as it touches on issues raised
         by Friedman. We don't intend a general survey; much will be left out. In
@@ -104,6 +115,6 @@ addr="friedman-1975">3</fr:link>) in the context of what we know today about the
             </fr:li>
             <fr:li>
                 logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+            </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="statman-1985.pdf">statman-1985.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/goldblatt-2006.xml b/_site/forest/goldblatt-2006.xml
index ff4e4d6..4f6291f 100644
--- a/_site/forest/goldblatt-2006.xml
+++ b/_site/forest/goldblatt-2006.xml
@@ -1,17 +1,402 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>486</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>668</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Bibliographical Note: This Dover edition, first published in 2006, is a
     slightly corrected unabridged republication of the revised (second) edition
     of the work originally published in 1984 by Elsevier Science Publishers
     B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
     "Studies in Logic and the Foundations of Mathematics." A new Preface and
     updated Bibliography have been specially prepared for this reprint."
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>669</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+type="local"
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
+        classifier and a natural numbers object."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+type="local"
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
+type="local"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course, if we ignore natural numbers objects for now.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-0001.xml b/_site/forest/hejohns-0001.xml
index 1139338..5910595 100644
--- a/_site/forest/hejohns-0001.xml
+++ b/_site/forest/hejohns-0001.xml
@@ -1,143 +1,179 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>819</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>930</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Here is Blackburn et all's account:
 </fr:p><fr:p>
     The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "The basic case":
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
         </fr:li>
         <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
         <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
 
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>.
     A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
     Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>817</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>932</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
     I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
     up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
+type="external"
+href="https://github.com/tomdjong/MGS-categorical-realizability">notes on
     realizability</fr:link>,
     according to Loris it's relevant to Markov Decision Processes, and we
     develop the view that it's a logician's view of fibrations.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>439</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Here is Blackburn et all's account:
 </fr:p><fr:p>
     The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "The basic case":
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
         </fr:li>
         <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
         <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
 
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>.
     A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
     Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-0002.xml b/_site/forest/hejohns-0002.xml
index b991eab..d0d2e4b 100644
--- a/_site/forest/hejohns-0002.xml
+++ b/_site/forest/hejohns-0002.xml
@@ -1,97 +1,129 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>811</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>922</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "The basic case":
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
         </fr:li>
         <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
         <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
 
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>809</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>924</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Here is Blackburn et all's account:
 </fr:p><fr:p>
     The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "The basic case":
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
         </fr:li>
         <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
         <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
 
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>.
     A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
     Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-0003.xml b/_site/forest/hejohns-0003.xml
index 15a3be2..f64c72c 100644
--- a/_site/forest/hejohns-0003.xml
+++ b/_site/forest/hejohns-0003.xml
@@ -1,49 +1,68 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>807</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>918</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
 
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>805</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>920</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "The basic case":
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
         </fr:li>
         <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
         <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
 
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-0004.xml b/_site/forest/hejohns-0004.xml
index 3eb0902..55b076b 100644
--- a/_site/forest/hejohns-0004.xml
+++ b/_site/forest/hejohns-0004.xml
@@ -1,77 +1,100 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>815</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>926</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
     Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>813</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>928</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Here is Blackburn et all's account:
 </fr:p><fr:p>
     The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "The basic case":
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
         </fr:li>
         <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
         <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
 
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>.
     A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
     Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-0005.xml b/_site/forest/hejohns-0005.xml
index 98c739c..3fc7cd4 100644
--- a/_site/forest/hejohns-0005.xml
+++ b/_site/forest/hejohns-0005.xml
@@ -1,82 +1,97 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>821</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>507</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
     I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
     up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
+type="external"
+href="https://github.com/tomdjong/MGS-categorical-realizability">notes on
     realizability</fr:link>,
     according to Loris it's relevant to Markov Decision Processes, and we
     develop the view that it's a logician's view of fibrations.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>439</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Here is Blackburn et all's account:
 </fr:p><fr:p>
     The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "The basic case":
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
         </fr:li>
         <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
         <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
 
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>.
     A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
     Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-0006.xml b/_site/forest/hejohns-0006.xml
index b939a23..6597a87 100644
--- a/_site/forest/hejohns-0006.xml
+++ b/_site/forest/hejohns-0006.xml
@@ -1,480 +1,673 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>713</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>880</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>882</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>711</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>883</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Mentioned in passing as <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
         <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
     This is conventionally left implicit, but we remark to clarify what we mean
     by the family of natural isomorphisms being natural.
 </fr:p><fr:p>
     NOTE: I couldn't get よ to render in the latex (diagrams).
     I'm not sure if it's a lualatex/dvi or font issue.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -482,480 +675,791 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
     so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>885</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+type="local"
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
+        Thus <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>887</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>705</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
+            <fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
+            <fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -963,347 +1467,306 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>707</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    However, unlike <fr:link
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>709</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>889</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -1311,39 +1774,8 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>704</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-0007.xml b/_site/forest/hejohns-0007.xml
index e470d41..4324204 100644
--- a/_site/forest/hejohns-0007.xml
+++ b/_site/forest/hejohns-0007.xml
@@ -1,318 +1,421 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>695</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>856</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Mentioned in passing as <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
         <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
     This is conventionally left implicit, but we remark to clarify what we mean
     by the family of natural isomorphisms being natural.
 </fr:p><fr:p>
     NOTE: I couldn't get よ to render in the latex (diagrams).
     I'm not sure if it's a lualatex/dvi or font issue.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -320,415 +423,586 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
     so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>858</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>693</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>859</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Mentioned in passing as <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
         <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
     This is conventionally left implicit, but we remark to clarify what we mean
     by the family of natural isomorphisms being natural.
 </fr:p><fr:p>
     NOTE: I couldn't get よ to render in the latex (diagrams).
     I'm not sure if it's a lualatex/dvi or font issue.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -736,613 +1010,768 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
     so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>691</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
-    </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>689</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>861</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>688</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>863</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+type="local"
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
+        Thus <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-0008.xml b/_site/forest/hejohns-0008.xml
index c27be33..cbb2704 100644
--- a/_site/forest/hejohns-0008.xml
+++ b/_site/forest/hejohns-0008.xml
@@ -1,404 +1,533 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>660</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>913</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Mentioned in passing as <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
         <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
     This is conventionally left implicit, but we remark to clarify what we mean
     by the family of natural isomorphisms being natural.
 </fr:p><fr:p>
     NOTE: I couldn't get よ to render in the latex (diagrams).
     I'm not sure if it's a lualatex/dvi or font issue.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -406,282 +535,361 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
     so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>659</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>915</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -690,4 +898,4 @@ root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:ta
         only on objects and the usual hom adjointness isomorphism implied the
         effect of the functors on homs was uniquely determined. Writing this down
         properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-0009.xml b/_site/forest/hejohns-0009.xml
index 35199fe..aec58dd 100644
--- a/_site/forest/hejohns-0009.xml
+++ b/_site/forest/hejohns-0009.xml
@@ -1,20 +1,21 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>657</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>739</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     What is the actual <fr:em>logic</fr:em> when doing a logical relation?
     Do you only actually need functors? Zeilberger's <fr:link
-href="millies-zeilberger-2015.pdf"
-type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
+type="external"
+href="millies-zeilberger-2015.pdf">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
     Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
 </fr:p><fr:p>
     Talked to Max about this on 5-9.
@@ -22,23 +23,31 @@ type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>m
     a hoare logic, and a implicational (eg FO) logic.
     I suppose one'd want to study a bit the relationship between the two.
     <fr:link
-href="millies-zeilberger-2015.xml"
 type="local"
-addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="millies-zeilberger-2015.xml"
+addr="millies-zeilberger-2015"
+title="Functors are Type Refinement Systems">paper</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>656</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>741</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -54,18 +63,20 @@ root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:taxon>Abstract</fr:ta
         present a few applications. As a larger case study, we revisit
         Reynolds’ paper on “The Meaning of Types” (2000), showing how the
         paper’s main results may be reconstructed along these lines.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex
+display="inline"><![CDATA[∀M]]></fr:tex> statements.
         Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
         functor into) the syntax, typically trivially constructed as a pullback
         (it's hard in general to concoct models with structure preserving maps
@@ -117,4 +128,4 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-z
     </fr:p><fr:p>
         Maybe I'm finally close to being able to understand what a Kripke
         logical relation is.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000A.xml b/_site/forest/hejohns-000A.xml
index 361adab..df62373 100644
--- a/_site/forest/hejohns-000A.xml
+++ b/_site/forest/hejohns-000A.xml
@@ -1,15 +1,16 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>654</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>718</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Fibered Right Adjoints are automatically fibered?
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000B.xml b/_site/forest/hejohns-000B.xml
index 053c27e..f0a800f 100644
--- a/_site/forest/hejohns-000B.xml
+++ b/_site/forest/hejohns-000B.xml
@@ -1,20 +1,22 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>566</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>725</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     While reading <fr:link
-href="von-neumann-1925.xml"
 type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
     "that's… the point"-- "realization" that first order logic is the
     meta-language of axiomatic mathematics, just as (it seems) logical
     frameworks are the meta-languages of (typically λ-calculus based)
@@ -23,26 +25,38 @@ addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had th
 </fr:p><fr:p>
     That is, (classical) FoL is often the setting in which axiomatic
     mathematical theories are formalized. Classic (no pun intended) examples
-    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
+    are <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{group} = \{e, ⋅, -⁻¹\}, Th(\textsf {groups}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{DLO} = \{≤\}, Th(\textsf {Dense Linear Orders}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{sets} = \{∈\}, ZFC)]]></fr:tex>.
     (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
 </fr:p><fr:p>
     "definite property"
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>727</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
-href="quine-1960.xml"
 type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
+href="quine-1960.xml"
+addr="quine-1960"
+title="Variables Explained Away">Variables Explained Away</fr:link>.
 </fr:p><fr:p>
     English translation in <fr:link
-href="van-heijenoort-1967.xml"
 type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="van-heijenoort-1967.xml"
+addr="van-heijenoort-1967"
+title="From Frege To Gödel">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000C.xml b/_site/forest/hejohns-000C.xml
index e36c807..343f936 100644
--- a/_site/forest/hejohns-000C.xml
+++ b/_site/forest/hejohns-000C.xml
@@ -1,26 +1,28 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>552</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Some basic questions, whose "elevator pitch" answers ought to roll off
         the tongue:
         <fr:ul><fr:li>
@@ -41,9 +43,10 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
                 Is it the same thing as "topos logic"?
             </fr:li></fr:ul>
         Following <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
         Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
@@ -70,37 +73,41 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sket
         complete the sentence".
         A truly FAQ-style answer compendium will have to come later-- after I've
         actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -114,65 +121,75 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -188,107 +205,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -303,148 +343,196 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -465,248 +553,377 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -714,7 +931,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -722,170 +940,267 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>546</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>538</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>540</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -915,8 +1230,11 @@ root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:ta
         and transformation of the thing. Doing this for "set theory" requires
         taking account of the experience that the main pairs of opposing
         tendencies in mathematics take the form of adjoint functors, and frees
-        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
-        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        us of the mathematically irrelevant traces (<fr:tex
+display="inline"><![CDATA[∈]]></fr:tex>) left behind by the
+        process of accumulating (<fr:tex
+display="inline"><![CDATA[∪]]></fr:tex>) the power set (<fr:tex
+display="inline"><![CDATA[\mathcal {P}]]></fr:tex>) at each stage of a
         metaphysical "construction". Further, experience with sheaves,
         permutation representations, algebraic spaces, etc., shows that a "set
         theory" for geometry should apply not only to abstract sets divorced
@@ -924,26 +1242,41 @@ root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:ta
         sets which do in fact develop along such parameters. For such sets,
         usually logic is "intuitionistic" (in its formal properties)
         usually the axiom of choice is false, and usually a set is not
-        determined by its points defined over <fr:tex>1</fr:tex> only.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+        determined by its points defined over <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> only.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     I have not read this, and have no idea what "contradictions" he's referring to.
     <fr:link
-href="lawvere-1970.pdf"
-type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="lawvere-1970.pdf">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>548</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>541</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>542</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>543</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -954,64 +1287,79 @@ root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Abstract</fr:ta
         remarks on a categorical view of the history of set theory, including a
         false history plausible from that point of view that would make it helpful
         to introduce toposes as a generalization from set theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="mclarty-1990b.pdf"
-type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="mclarty-1990b.pdf">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>549</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>544</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
-href="ronnie-chen.xml"
 type="local"
-addr="ronnie-chen">Ronnie Chen</fr:link>,
+href="ronnie-chen.xml"
+addr="ronnie-chen"
+title="Ronnie Chen">Ronnie Chen</fr:link>,
     where the viewpoint of "generalized algebraic semantics" is developed.
 </fr:p><fr:p><fr:link
-href="marquis-reyes-2012.pdf"
-type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="marquis-reyes-2012.pdf">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>550</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>545</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>546</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>547</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
         computation and the geometric side of topos theory. We begin with a brief
         outline of the history and basic concepts of topos theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
         the history of topos theory, the relation between Grothendieck and
         Elementary topoi, and the relation between geometric and logical
         morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="blass-1988.pdf">blass-1988.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000D.xml b/_site/forest/hejohns-000D.xml
index 5bc9ba7..26c2fa9 100644
--- a/_site/forest/hejohns-000D.xml
+++ b/_site/forest/hejohns-000D.xml
@@ -1,220 +1,292 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>652</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>899</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -222,322 +294,433 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>650</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>901</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Mentioned in passing as <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
         <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
     This is conventionally left implicit, but we remark to clarify what we mean
     by the family of natural isomorphisms being natural.
 </fr:p><fr:p>
     NOTE: I couldn't get よ to render in the latex (diagrams).
     I'm not sure if it's a lualatex/dvi or font issue.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -545,395 +728,512 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
     so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>642</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>644</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>646</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>648</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>903</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
-    </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>640</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>905</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>907</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>909</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+type="local"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>911</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+type="local"
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
+        Thus <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000E.xml b/_site/forest/hejohns-000E.xml
index 280d701..cf5251d 100644
--- a/_site/forest/hejohns-000E.xml
+++ b/_site/forest/hejohns-000E.xml
@@ -1,653 +1,890 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>618</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>865</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>867</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>616</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>868</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Mentioned in passing as <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
         <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
     This is conventionally left implicit, but we remark to clarify what we mean
     by the family of natural isomorphisms being natural.
 </fr:p><fr:p>
     NOTE: I couldn't get よ to render in the latex (diagrams).
     I'm not sure if it's a lualatex/dvi or font issue.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -655,516 +892,804 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
     so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>870</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
+type="local"
+href="unstable-214.xml"
+addr="#214">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>872</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="unstable-214.xml"
+addr="#214">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>874</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
+            <fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>610</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
+            <fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -1172,322 +1697,378 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
+type="local"
 href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    However, unlike <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>876</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>878</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -1495,97 +2076,8 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000F.xml b/_site/forest/hejohns-000F.xml
index 19c53c4..617f130 100644
--- a/_site/forest/hejohns-000F.xml
+++ b/_site/forest/hejohns-000F.xml
@@ -1,30 +1,33 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>492</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -45,27 +48,86 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="blass-1988.pdf">blass-1988.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Some basic questions, whose "elevator pitch" answers ought to roll off
         the tongue:
         <fr:ul><fr:li>
@@ -86,9 +148,10 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
                 Is it the same thing as "topos logic"?
             </fr:li></fr:ul>
         Following <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
         Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
@@ -115,37 +178,41 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sket
         complete the sentence".
         A truly FAQ-style answer compendium will have to come later-- after I've
         actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -159,65 +226,75 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -233,107 +310,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -348,148 +448,196 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -510,248 +658,377 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -759,7 +1036,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -767,374 +1045,590 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>489</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>416</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -1142,7 +1636,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -1150,174 +1645,211 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>488</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000G.xml b/_site/forest/hejohns-000G.xml
index 71716d8..5d9c3d4 100644
--- a/_site/forest/hejohns-000G.xml
+++ b/_site/forest/hejohns-000G.xml
@@ -1,59 +1,68 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>528</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -69,107 +78,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -184,144 +216,291 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>602</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>604</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>605</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>606</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>526</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="mclarty-1990b.pdf">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>607</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>608</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>609</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Some basic questions, whose "elevator pitch" answers ought to roll off
         the tongue:
         <fr:ul><fr:li>
@@ -342,9 +521,10 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
                 Is it the same thing as "topos logic"?
             </fr:li></fr:ul>
         Following <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
         Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
@@ -371,37 +551,41 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sket
         complete the sentence".
         A truly FAQ-style answer compendium will have to come later-- after I've
         actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -415,65 +599,75 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -489,107 +683,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -604,148 +821,196 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -766,248 +1031,377 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -1015,7 +1409,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -1023,396 +1418,590 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>524</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
-type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
-            <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
-                </fr:li>
-                <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
-                </fr:li>
-                <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>611</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -1420,7 +2009,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -1428,172 +2018,259 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>520</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The view that toposes originated as generalized set theory is a figment of
-        set theoretically educated common sense. This false history obstructs
-        understanding of category theory and especially of categorical foundations
-        for mathematics. Problems in geometry, topology, and related algebra led to
-        categories and toposes. Elementary toposes arose when Lawvere's interest in
-        the foundations of physics and Tierney's in the foundations of topology led
-        both to study Grothendieck's foundations for algebraic geometry. I end with
-        remarks on a categorical view of the history of set theory, including a
-        false history plausible from that point of view that would make it helpful
-        to introduce toposes as a generalization from set theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="mclarty-1990b.pdf"
-type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>522</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>612</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title><fr:link
+type="local"
+href="unstable-199.xml"
+addr="#199"
+title="What is an elementary topos? › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory"><fr:link
+type="local"
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">What is an elementary topos?</fr:link> › Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000H.xml b/_site/forest/hejohns-000H.xml
index 3f3ab56..a37d2f2 100644
--- a/_site/forest/hejohns-000H.xml
+++ b/_site/forest/hejohns-000H.xml
@@ -1,26 +1,29 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>535</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>671</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -34,32 +37,163 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>673</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex
+display="inline"><![CDATA[∈]]></fr:tex>) left behind by the
+        process of accumulating (<fr:tex
+display="inline"><![CDATA[∪]]></fr:tex>) the power set (<fr:tex
+display="inline"><![CDATA[\mathcal {P}]]></fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> only.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+type="external"
+href="lawvere-1970.pdf">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>674</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="ronnie-chen.xml"
+addr="ronnie-chen"
+title="Ronnie Chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+type="external"
+href="marquis-reyes-2012.pdf">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>675</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="blass-1988.pdf">blass-1988.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>676</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Some basic questions, whose "elevator pitch" answers ought to roll off
         the tongue:
         <fr:ul><fr:li>
@@ -80,9 +214,10 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
                 Is it the same thing as "topos logic"?
             </fr:li></fr:ul>
         Following <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
         Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
@@ -109,37 +244,41 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sket
         complete the sentence".
         A truly FAQ-style answer compendium will have to come later-- after I've
         actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -153,65 +292,75 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -227,107 +376,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -342,148 +514,196 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -504,248 +724,377 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -753,7 +1102,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -761,248 +1111,219 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>530</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The unity of opposites in the title is essentially that between logic
-        and geometry, and there are compelling reasons for maintaining that
-        geometry is the leading aspect. At the same lime, in the present joint
-        work with Myles Tierney there are important influences in the other
-        direction: a Grothendieck "topology" appears most naturally as a
-        modal operator, of the nature "it is locally the case that", the
-        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
-        apply to families of geometrical objects rather than to propositional
-        functions, and an important technique is to lift constructions first
-        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
-        topos. We first sum up the principal contradictions of the
-        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
-        adjoint functors, significantly generalizing the theory to free it of
-        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
-        enabling one to claim that in a sense logic is a special case of
-        geometry). The method thus developing is then applied to intrinsically
-        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
-        the independence of the continuum hypothesis free of any use of
-        transfinite induction. The second application of the method outlined
-        here is an intrinsic geometric construction of the Chevalley-Hakim
-        global spectrum of a ringed topos free of any choice of a "site of
-        definition".
-    </fr:p><fr:p>
-        When the main contradictions of a thing have been found,
-        the scientific procedure is to summarize them in slogans which one then
-        constantly uses as an ideological weapon for the further development
-        and transformation of the thing. Doing this for "set theory" requires
-        taking account of the experience that the main pairs of opposing
-        tendencies in mathematics take the form of adjoint functors, and frees
-        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
-        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
-        metaphysical "construction". Further, experience with sheaves,
-        permutation representations, algebraic spaces, etc., shows that a "set
-        theory" for geometry should apply not only to abstract sets divorced
-        from time, space, ring of definition, etc., but also to more general
-        sets which do in fact develop along such parameters. For such sets,
-        usually logic is "intuitionistic" (in its formal properties)
-        usually the axiom of choice is false, and usually a set is not
-        determined by its points defined over <fr:tex>1</fr:tex> only.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    I have not read this, and have no idea what "contradictions" he's referring to.
-    <fr:link
-href="lawvere-1970.pdf"
-type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I have not read this. This might be a good source, aside from <fr:link
-href="ronnie-chen.xml"
-type="local"
-addr="ronnie-chen">Ronnie Chen</fr:link>,
-    where the viewpoint of "generalized algebraic semantics" is developed.
-</fr:p><fr:p><fr:link
-href="marquis-reyes-2012.pdf"
-type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>532</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000I.xml b/_site/forest/hejohns-000I.xml
index 99283d5..16e45cf 100644
--- a/_site/forest/hejohns-000I.xml
+++ b/_site/forest/hejohns-000I.xml
@@ -1,89 +1,126 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>478</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>631</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>632</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>633</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -99,107 +136,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -214,146 +274,195 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>475</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000K</fr:addr><fr:route>hejohns-000K.xml</fr:route><fr:title>Global element</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A <fr:em>global element</fr:em> of an object <fr:tex>A ∈  \mathcal {C}</fr:tex> is <fr:em>just</fr:em> a
-    morphism <fr:tex>e : 1 → A</fr:tex>.
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>635</fr:anchor><fr:addr>hejohns-000K</fr:addr><fr:route>hejohns-000K.xml</fr:route><fr:title>Global element</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A <fr:em>global element</fr:em> of an object <fr:tex
+display="inline"><![CDATA[A ∈ \mathcal {C}]]></fr:tex> is <fr:em>just</fr:em> a
+    morphism <fr:tex
+display="inline"><![CDATA[e : 1 → A]]></fr:tex>.
 </fr:p><fr:p>
     From <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 href="hejohns-000L.xml"
-type="local"
-addr="hejohns-000L">generalized element</fr:link>.
-</fr:p></fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+addr="hejohns-000L"
+title="Generalized element">generalized element</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000J.xml b/_site/forest/hejohns-000J.xml
index e37e5cd..4ea0777 100644
--- a/_site/forest/hejohns-000J.xml
+++ b/_site/forest/hejohns-000J.xml
@@ -1,247 +1,376 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>473</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>623</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -249,7 +378,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -257,162 +387,279 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>624</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>625</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>626</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>471</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>627</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Some basic questions, whose "elevator pitch" answers ought to roll off
         the tongue:
         <fr:ul><fr:li>
@@ -433,9 +680,10 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
                 Is it the same thing as "topos logic"?
             </fr:li></fr:ul>
         Following <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
         Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
@@ -462,37 +710,41 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sket
         complete the sentence".
         A truly FAQ-style answer compendium will have to come later-- after I've
         actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -506,65 +758,75 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -580,107 +842,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -695,148 +980,196 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -857,248 +1190,377 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -1106,7 +1568,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -1114,146 +1577,219 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000K.xml b/_site/forest/hejohns-000K.xml
index 97b629d..441de58 100644
--- a/_site/forest/hejohns-000K.xml
+++ b/_site/forest/hejohns-000K.xml
@@ -1,100 +1,146 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>469</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000K</fr:addr><fr:route>hejohns-000K.xml</fr:route><fr:title>Global element</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A <fr:em>global element</fr:em> of an object <fr:tex>A ∈  \mathcal {C}</fr:tex> is <fr:em>just</fr:em> a
-    morphism <fr:tex>e : 1 → A</fr:tex>.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>618</fr:anchor><fr:addr>hejohns-000K</fr:addr><fr:route>hejohns-000K.xml</fr:route><fr:title>Global element</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A <fr:em>global element</fr:em> of an object <fr:tex
+display="inline"><![CDATA[A ∈ \mathcal {C}]]></fr:tex> is <fr:em>just</fr:em> a
+    morphism <fr:tex
+display="inline"><![CDATA[e : 1 → A]]></fr:tex>.
 </fr:p><fr:p>
     From <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 href="hejohns-000L.xml"
-type="local"
-addr="hejohns-000L">generalized element</fr:link>.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns-000L"
+title="Generalized element">generalized element</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>619</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>468</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000L</fr:addr><fr:route>hejohns-000L.xml</fr:route><fr:title>Generalized element</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A <fr:em>generalized element</fr:em> (alt. local element) of an object <fr:tex>A ∈      \mathcal {C}</fr:tex> is <fr:em>just</fr:em> a morphism <fr:tex>e : B → A</fr:tex>, for some <fr:tex>B</fr:tex>. Ie, a
-    generalized element of <fr:tex>A</fr:tex> is <fr:em>just</fr:em> an object of the slice over <fr:tex>A</fr:tex>.
-</fr:p><fr:p>
-    There's supposed to be some Kripke-style interpretation of these as
-    elements defined at points in time, but TODO.
-</fr:p><fr:p>
-    From <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
-href="hejohns-000K.xml"
-type="local"
-addr="hejohns-000K">global element</fr:link>.
-</fr:p></fr:mainmatter></fr:tree></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>466</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000L</fr:addr><fr:route>hejohns-000L.xml</fr:route><fr:title>Generalized element</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A <fr:em>generalized element</fr:em> (alt. local element) of an object <fr:tex>A ∈      \mathcal {C}</fr:tex> is <fr:em>just</fr:em> a morphism <fr:tex>e : B → A</fr:tex>, for some <fr:tex>B</fr:tex>. Ie, a
-    generalized element of <fr:tex>A</fr:tex> is <fr:em>just</fr:em> an object of the slice over <fr:tex>A</fr:tex>.
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:addr>hejohns-000L</fr:addr><fr:route>hejohns-000L.xml</fr:route><fr:title>Generalized element</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A <fr:em>generalized element</fr:em> (alt. local element) of an object <fr:tex
+display="inline"><![CDATA[A ∈     \mathcal {C}]]></fr:tex> is <fr:em>just</fr:em> a morphism <fr:tex
+display="inline"><![CDATA[e : B → A]]></fr:tex>, for some <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. Ie, a
+    generalized element of <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is <fr:em>just</fr:em> an object of the slice over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
 </fr:p><fr:p>
     There's supposed to be some Kripke-style interpretation of these as
     elements defined at points in time, but TODO.
 </fr:p><fr:p>
     From <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 href="hejohns-000K.xml"
-type="local"
-addr="hejohns-000K">global element</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns-000K"
+title="Global element">global element</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>467</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>621</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>465</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>622</fr:anchor><fr:addr>hejohns-000L</fr:addr><fr:route>hejohns-000L.xml</fr:route><fr:title>Generalized element</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A <fr:em>generalized element</fr:em> (alt. local element) of an object <fr:tex
+display="inline"><![CDATA[A ∈     \mathcal {C}]]></fr:tex> is <fr:em>just</fr:em> a morphism <fr:tex
+display="inline"><![CDATA[e : B → A]]></fr:tex>, for some <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. Ie, a
+    generalized element of <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is <fr:em>just</fr:em> an object of the slice over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+</fr:p><fr:p>
+    There's supposed to be some Kripke-style interpretation of these as
+    elements defined at points in time, but TODO.
+</fr:p><fr:p>
+    From <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
+type="local"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global element</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000L.xml b/_site/forest/hejohns-000L.xml
index 9b93fe2..0da78f6 100644
--- a/_site/forest/hejohns-000L.xml
+++ b/_site/forest/hejohns-000L.xml
@@ -1,67 +1,103 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>464</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000L</fr:addr><fr:route>hejohns-000L.xml</fr:route><fr:title>Generalized element</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A <fr:em>generalized element</fr:em> (alt. local element) of an object <fr:tex>A ∈      \mathcal {C}</fr:tex> is <fr:em>just</fr:em> a morphism <fr:tex>e : B → A</fr:tex>, for some <fr:tex>B</fr:tex>. Ie, a
-    generalized element of <fr:tex>A</fr:tex> is <fr:em>just</fr:em> an object of the slice over <fr:tex>A</fr:tex>.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>614</fr:anchor><fr:addr>hejohns-000L</fr:addr><fr:route>hejohns-000L.xml</fr:route><fr:title>Generalized element</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A <fr:em>generalized element</fr:em> (alt. local element) of an object <fr:tex
+display="inline"><![CDATA[A ∈     \mathcal {C}]]></fr:tex> is <fr:em>just</fr:em> a morphism <fr:tex
+display="inline"><![CDATA[e : B → A]]></fr:tex>, for some <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. Ie, a
+    generalized element of <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is <fr:em>just</fr:em> an object of the slice over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
 </fr:p><fr:p>
     There's supposed to be some Kripke-style interpretation of these as
     elements defined at points in time, but TODO.
 </fr:p><fr:p>
     From <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 href="hejohns-000K.xml"
-type="local"
-addr="hejohns-000K">global element</fr:link>.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns-000K"
+title="Global element">global element</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>615</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>463</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000K</fr:addr><fr:route>hejohns-000K.xml</fr:route><fr:title>Global element</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A <fr:em>global element</fr:em> of an object <fr:tex>A ∈  \mathcal {C}</fr:tex> is <fr:em>just</fr:em> a
-    morphism <fr:tex>e : 1 → A</fr:tex>.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>616</fr:anchor><fr:addr>hejohns-000K</fr:addr><fr:route>hejohns-000K.xml</fr:route><fr:title>Global element</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A <fr:em>global element</fr:em> of an object <fr:tex
+display="inline"><![CDATA[A ∈ \mathcal {C}]]></fr:tex> is <fr:em>just</fr:em> a
+    morphism <fr:tex
+display="inline"><![CDATA[e : 1 → A]]></fr:tex>.
 </fr:p><fr:p>
     From <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 href="hejohns-000L.xml"
-type="local"
-addr="hejohns-000L">generalized element</fr:link>.
-</fr:p></fr:mainmatter></fr:tree></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns-000L"
+title="Generalized element">generalized element</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>462</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000K</fr:addr><fr:route>hejohns-000K.xml</fr:route><fr:title>Global element</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A <fr:em>global element</fr:em> of an object <fr:tex>A ∈  \mathcal {C}</fr:tex> is <fr:em>just</fr:em> a
-    morphism <fr:tex>e : 1 → A</fr:tex>.
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>617</fr:anchor><fr:addr>hejohns-000K</fr:addr><fr:route>hejohns-000K.xml</fr:route><fr:title>Global element</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A <fr:em>global element</fr:em> of an object <fr:tex
+display="inline"><![CDATA[A ∈ \mathcal {C}]]></fr:tex> is <fr:em>just</fr:em> a
+    morphism <fr:tex
+display="inline"><![CDATA[e : 1 → A]]></fr:tex>.
 </fr:p><fr:p>
     From <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
 href="hejohns-000L.xml"
-type="local"
-addr="hejohns-000L">generalized element</fr:link>.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>461</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+addr="hejohns-000L"
+title="Generalized element">generalized element</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000M.xml b/_site/forest/hejohns-000M.xml
index 049fef3..6fb8388 100644
--- a/_site/forest/hejohns-000M.xml
+++ b/_site/forest/hejohns-000M.xml
@@ -1,238 +1,367 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>460</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>491</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -240,7 +369,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -248,375 +378,619 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>492</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>493</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>494</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>495</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -624,7 +998,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -632,175 +1007,256 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>ericgio</fr:addr><fr:route>ericgio.xml</fr:route><fr:title>Eric Giovannini</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://public.websites.umich.edu/~ericgio/index.html"
-type="external">https://public.websites.umich.edu/~ericgio/index.html</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>455</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>496</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -821,60 +1277,78 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>498</fr:anchor><fr:addr>ericgio</fr:addr><fr:route>ericgio.xml</fr:route><fr:title>Eric Giovannini</fr:title><fr:taxon>Person</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://public.websites.umich.edu/~ericgio/index.html">https://public.websites.umich.edu/~ericgio/index.html</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>499</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -890,107 +1364,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -1005,128 +1502,166 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>453</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000N.xml b/_site/forest/hejohns-000N.xml
index ea82b30..568d880 100644
--- a/_site/forest/hejohns-000N.xml
+++ b/_site/forest/hejohns-000N.xml
@@ -1,23 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>451</fr:anchor><fr:addr>hejohns-000N</fr:addr><fr:route>hejohns-000N.xml</fr:route><fr:title>L (Language)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>558</fr:anchor><fr:addr>hejohns-000N</fr:addr><fr:route>hejohns-000N.xml</fr:route><fr:title>L (Language)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     I vaguely recall reading that the tradition of isolating the non-logical
     symbols (constants, functions, relations) of a first order logic for a
-    language <fr:tex>\mathcal {L}</fr:tex>, and referring to suitable models as
-    <fr:tex>\mathcal {L}</fr:tex>-structures, was popularized by Carnap, and maybe in
+    language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, and referring to suitable models as
+    <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>-structures, was popularized by Carnap, and maybe in
     particular by <fr:em>The Logical Syntax of Language</fr:em>.
 </fr:p><fr:p>
     I am unable to track down a source for this now. I'll have to check <fr:em>The
     Syntax</fr:em> sometime. I can say at least <fr:em>Meaning and Necessity</fr:em> does
     <fr:em>not</fr:em> make use of this terminology in the same way.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000O.xml b/_site/forest/hejohns-000O.xml
index f984005..b2997e2 100644
--- a/_site/forest/hejohns-000O.xml
+++ b/_site/forest/hejohns-000O.xml
@@ -1,186 +1,128 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>447</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Basic information</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    I am a master's student in Computer Science and Engineering at the
-    University of Michigan.
-    I'm interested in logic and related areas of computer science, particularly
-    programming languages.
-</fr:p><fr:p>
-    This is my <fr:link
-href="https://www.jonmsterling.com/jms-005P.xml"
-type="external">forest</fr:link>.
-</fr:p><fr:ul><fr:li>
-                email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li>
-    <fr:li>
-        photo:
-        <fr:p><html:img
-xmlns:html="http://www.w3.org/1999/xhtml"
-src="hejohns.jpeg"
-width="150"
-height="150"></html:img></fr:p></fr:li>
-    <fr:li>
-        cv: <fr:link
-href="cv.pdf"
-type="external">cv.pdf</fr:link> (TODO)
-    </fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>275</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>548</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Basic information</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>271</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Short About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     I am a master's student in Computer Science and Engineering at the
     University of Michigan.
     I'm interested in logic and related areas of computer science, particularly
     programming languages.
 </fr:p><fr:p>
     This is my <fr:link
-href="https://www.jonmsterling.com/jms-005P.xml"
-type="external">forest</fr:link>.
+type="external"
+href="https://www.jonmsterling.com/jms-005P.xml">forest</fr:link>.
 </fr:p><fr:ul><fr:li>
                 email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li>
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble><![CDATA[]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[hejohns@umich.edu]]></fr:embedded-tex-body></fr:embedded-tex></fr:li>
     <fr:li>
         photo:
         <fr:p><html:img
 xmlns:html="http://www.w3.org/1999/xhtml"
 src="hejohns.jpeg"
 width="150"
-height="150"></html:img></fr:p></fr:li>
+height="150" /></fr:p></fr:li>
     <fr:li>
         cv: <fr:link
-href="cv.pdf"
-type="external">cv.pdf</fr:link> (TODO)
-    </fr:li></fr:ul></fr:mainmatter></fr:tree>
-<fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+type="external"
+href="cv.pdf">cv.pdf</fr:link> (TODO)
+    </fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>273</fr:anchor><fr:title>Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        My screen name is <fr:code>hejohns</fr:code>, eg <fr:link
-href="https://github.com/hejohns"
-type="external">https://github.com/hejohns</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="true"><fr:frontmatter><fr:anchor>445</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:contributor><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors><fr:meta
+expanded="false"
+root="true"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>550</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:meta
 name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>277</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>219</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     I am a master's student in Computer Science and Engineering at the
     University of Michigan.
     I'm interested in logic and related areas of computer science, particularly
     programming languages.
 </fr:p><fr:p>
     This is my <fr:link
-href="https://www.jonmsterling.com/jms-005P.xml"
-type="external">forest</fr:link>.
+type="external"
+href="https://www.jonmsterling.com/jms-005P.xml">forest</fr:link>.
 </fr:p><fr:ul><fr:li>
                 email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li>
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble><![CDATA[]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[hejohns@umich.edu]]></fr:embedded-tex-body></fr:embedded-tex></fr:li>
     <fr:li>
         photo:
         <fr:p><html:img
 xmlns:html="http://www.w3.org/1999/xhtml"
 src="hejohns.jpeg"
 width="150"
-height="150"></html:img></fr:p></fr:li>
+height="150" /></fr:p></fr:li>
     <fr:li>
         cv: <fr:link
-href="cv.pdf"
-type="external">cv.pdf</fr:link> (TODO)
-    </fr:li></fr:ul></fr:mainmatter></fr:tree>
+type="external"
+href="cv.pdf">cv.pdf</fr:link> (TODO)
+    </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree>
     <fr:p>
         For more information about me, see <fr:link
-href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link>.
     </fr:p>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="true"
-show-heading="false"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>357</fr:anchor></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>311</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>223</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Some basic questions, whose "elevator pitch" answers ought to roll off
         the tongue:
         <fr:ul><fr:li>
@@ -201,9 +143,10 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
                 Is it the same thing as "topos logic"?
             </fr:li></fr:ul>
         Following <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
         Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
@@ -230,37 +173,41 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sket
         complete the sentence".
         A truly FAQ-style answer compendium will have to come later-- after I've
         actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -274,65 +221,75 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -348,107 +305,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -463,148 +443,196 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -625,248 +653,377 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -874,7 +1031,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -882,540 +1040,749 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
-    </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
+    </fr:p><fr:p>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>270</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Mentioned in passing as <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
         <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
     This is conventionally left implicit, but we remark to clarify what we mean
     by the family of natural isomorphisms being natural.
 </fr:p><fr:p>
     NOTE: I couldn't get よ to render in the latex (diagrams).
     I'm not sure if it's a lualatex/dvi or font issue.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -1423,376 +1790,413 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
     so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="true"
-show-heading="false"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
     course text(s). I haven't used it much for reading either, owing to the
     large amount of new notation one has to refresh every time you open the
     book, but it's a solid reference book.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
-    title, and Makkai completeness.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    "Bibliographical Note: This Dover edition, first published in 2006, is a
-    slightly corrected unabridged republication of the revised (second) edition
-    of the work originally published in 1984 by Elsevier Science Publishers
-    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
-    "Studies in Logic and the Foundations of Mathematics." A new Preface and
-    updated Bibliography have been specially prepared for this reprint."
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="mclarty-1990a.pdf"
-type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -1822,8 +2226,11 @@ root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:ta
         and transformation of the thing. Doing this for "set theory" requires
         taking account of the experience that the main pairs of opposing
         tendencies in mathematics take the form of adjoint functors, and frees
-        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
-        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        us of the mathematically irrelevant traces (<fr:tex
+display="inline"><![CDATA[∈]]></fr:tex>) left behind by the
+        process of accumulating (<fr:tex
+display="inline"><![CDATA[∪]]></fr:tex>) the power set (<fr:tex
+display="inline"><![CDATA[\mathcal {P}]]></fr:tex>) at each stage of a
         metaphysical "construction". Further, experience with sheaves,
         permutation representations, algebraic spaces, etc., shows that a "set
         theory" for geometry should apply not only to abstract sets divorced
@@ -1831,52 +2238,41 @@ root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:ta
         sets which do in fact develop along such parameters. For such sets,
         usually logic is "intuitionistic" (in its formal properties)
         usually the axiom of choice is false, and usually a set is not
-        determined by its points defined over <fr:tex>1</fr:tex> only.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+        determined by its points defined over <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> only.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     I have not read this, and have no idea what "contradictions" he's referring to.
     <fr:link
-href="lawvere-1970.pdf"
-type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="lawvere-1970.pdf">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -1887,188 +2283,170 @@ root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Abstract</fr:ta
         remarks on a categorical view of the history of set theory, including a
         false history plausible from that point of view that would make it helpful
         to introduce toposes as a generalization from set theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="mclarty-1990b.pdf"
-type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="mclarty-1990b.pdf">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
-href="ronnie-chen.xml"
 type="local"
-addr="ronnie-chen">Ronnie Chen</fr:link>,
+href="ronnie-chen.xml"
+addr="ronnie-chen"
+title="Ronnie Chen">Ronnie Chen</fr:link>,
     where the viewpoint of "generalized algebraic semantics" is developed.
 </fr:p><fr:p><fr:link
-href="marquis-reyes-2012.pdf"
-type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="marquis-reyes-2012.pdf">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+type="local"
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">Equality Between Functionals</fr:link> and <fr:link
+type="local"
+href="statman-1985.xml"
+addr="statman-1985"
+title="Equality Between Functionals Revisited">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+type="external"
+href="https://proofassistants.stackexchange.com/a/1177">stack overflow</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    One of the early NBG papers. Maybe "the" NBG paper.
-    My notes on this are currently in <fr:link
-href="quine-1960.xml"
-type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
-</fr:p><fr:p>
-    English translation in <fr:link
-href="van-heijenoort-1967.xml"
-type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex
+display="inline"><![CDATA[⊢ s = t]]></fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+type="external"
+href="friedman-1975.pdf">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
-    this list. Whatever else you might think about Quine, he was a terrific
-    explainer, and this is the best introduction to the subject I know."
-</fr:p><fr:p>
-    (Yes, that 50 should be a 60(b).)
-    Here is the followup I wrote:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
     </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="statman-1985.pdf">statman-1985.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -2089,14 +2467,14 @@ root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Abstract</fr:ta
         stuck.  To achieve self-interpretation and show Jones-optimality, we
         relate this exemplar of higher-rank and higher-kind polymorphism to
         plugging a term into a context of let-polymorphic bindings.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -2118,21 +2496,116 @@ root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Notes</fr:taxon
         that it is the free syntax generated by the encoded rules/initial
         algebra/inductive dataype.
         But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex
+display="inline"><![CDATA[∀M]]></fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -2146,14 +2619,14 @@ root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:ta
         computability</fr:strong>, logical relations are manipulated as if they were sets,
         making the essence of many complex logical relations arguments
         accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -2166,37 +2639,236 @@ root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</f
         and I just don't have some context, or it's actually presented almost
         mystically.
         The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+href="carette-kiselyov-shan-2007.xml"
+addr="carette-kiselyov-shan-2007"
+title="Finally Tagless, Partially Evaluated">finally tagless encodings</fr:link>
         and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
+type="external"
+href="hoffman-1999">a justification for hoas</fr:link> doesn't seem super
         on-the-point, but if it's mentioned it must be for good reason…
     </fr:p><fr:p>
         Again, quotes are air-quotes, not actual quotes.
         <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="sterling-2022.pdf">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+type="external"
+href="blass-2003.pdf">blass-2003.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="blass-1988.pdf">blass-1988.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    One of the early NBG papers. Maybe "the" NBG paper.
+    My notes on this are currently in <fr:link
+type="local"
+href="quine-1960.xml"
+addr="quine-1960"
+title="Variables Explained Away">Variables Explained Away</fr:link>.
+</fr:p><fr:p>
+    English translation in <fr:link
+type="local"
+href="van-heijenoort-1967.xml"
+addr="van-heijenoort-1967"
+title="From Frege To Gödel">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A+. I should get a second copy to gift whenever needed.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
+    this list. Whatever else you might think about Quine, he was a terrific
+    explainer, and this is the best introduction to the subject I know."
+</fr:p><fr:p>
+    (Yes, that 50 should be a 60(b).)
+    Here is the followup I wrote:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
+    </fr:p><fr:p>
+        I was surprised by the mention of von Neumann 1925 in this snippet:
+    </fr:p><fr:p>
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
+    </fr:p><fr:p>
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+type="external"
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:ul><fr:li><fr:link
+type="external"
+href="quine-1960.pdf">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
+type="local"
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
         sympathy to this line of thought, I’m skeptical of the unspoken assumption
         that there’s an underlying concept of a ‘foundation’ up for analysis, that
         this analysis would properly guide our assessment of the various
@@ -2207,7 +2879,7 @@ root="false"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:taxon>Abstract</fr:ta
         done. What matters is these jobs we want our theories to do and how well
         they do them. Whether any or all of them, jobs or theories, deserves to be
         called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     A nice article that can be briefly summarized as
     <fr:ul><fr:li>
             There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
@@ -2241,475 +2913,234 @@ root="false"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:taxon>Abstract</fr:ta
 </fr:p><fr:p>
     I've used quotes here as air quotes, not actual quotes of Maddy.
     <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="maddy-2019.pdf">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
-    </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
-        <fr:ul><fr:li>
-                completness theorems,
-            </fr:li>
-            <fr:li>
-                the solvability of higher type functional equations, and
-            </fr:li>
-            <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
+type="external"
+href="zeilberger">zeilberger</fr:link> quotes on his work on
         polarity.
     </fr:p><fr:p>
         What's the relationship between harmony-stability and Pfenning's local
         soundness and completeness?
         They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The standard reading of type theory through the lens of category theory
-        is based on the idea of viewing a type system as a category of
-        well-typed terms. We propose a basic revision of this reading: rather
-        than interpreting type systems as categories, we describe them as
-        functors from a category of typing derivations to a category of
-        underlying terms. Then, turning this around, we explain how in fact any
-        functor gives rise to a generalized type system, with an abstract
-        notion of typing judgment, typing derivations and typing rules. This
-        leads to a purely categorical reformulation of various natural classes
-        of type systems as natural classes of functors. The main purpose of
-        this paper is to describe the general framework (which can also be seen
-        as providing a categorical analysis of refinement types), and to
-        present a few applications. As a larger case study, we revisit
-        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
-        paper’s main results may be reconstructed along these lines.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
-    </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
-    </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
-    </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
-    </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
-    </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
-    </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        TODO: not a good selection
-    </fr:p><fr:ul><fr:li><fr:link
-href="https://github.com/eecs490/gradescope-utils"
-type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Contributing to <fr:link
+type="external"
+href="https://github.com/maxsnew/cubical-categorical-logic">Cubical Categorical Logic</fr:link>, "cubical agda formalizations of different approaches to categorical semantics of type theories"
         </fr:li>
         <fr:li><fr:link
-href="https://github.com/hejohns/.rc/tree/master/home/bin"
-type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://mplse-reading-group.github.io/"
-type="external">https://mplse-reading-group.github.io/</fr:link></fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+type="external"
+href="https://mplse-reading-group.github.io/">https://mplse-reading-group.github.io/</fr:link></fr:li>
         <fr:li><fr:link
-href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
-type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
-href="../posts.html"
-type="external">Jekyll Blog</fr:link></fr:mainmatter></fr:tree>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
+type="external"
+href="../posts.html">Jekyll Blog</fr:link></fr:mainmatter><fr:backmatter /></fr:tree>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="true"
-show-heading="false"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>443</fr:anchor></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html"
-type="external">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
-href="stack-overflow/every-topos-is-a-regular-category.html"
-type="external">stack-overflow/every-topos-is-a-regular-category.html</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p /></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
-    title, and Makkai completeness.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
+type="external"
+href="stack-overflow/every-topos-is-a-regular-category.html">stack-overflow/every-topos-is-a-regular-category.html</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+type="external"
+href="https://github.com/tomdjong/MGS-categorical-realizability">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The unity of opposites in the title is essentially that between logic
-        and geometry, and there are compelling reasons for maintaining that
-        geometry is the leading aspect. At the same lime, in the present joint
-        work with Myles Tierney there are important influences in the other
-        direction: a Grothendieck "topology" appears most naturally as a
-        modal operator, of the nature "it is locally the case that", the
-        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
-        apply to families of geometrical objects rather than to propositional
-        functions, and an important technique is to lift constructions first
-        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
-        topos. We first sum up the principal contradictions of the
-        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
-        adjoint functors, significantly generalizing the theory to free it of
-        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
-        enabling one to claim that in a sense logic is a special case of
-        geometry). The method thus developing is then applied to intrinsically
-        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
-        the independence of the continuum hypothesis free of any use of
-        transfinite induction. The second application of the method outlined
-        here is an intrinsic geometric construction of the Chevalley-Hakim
-        global spectrum of a ringed topos free of any choice of a "site of
-        definition".
-    </fr:p><fr:p>
-        When the main contradictions of a thing have been found,
-        the scientific procedure is to summarize them in slogans which one then
-        constantly uses as an ideological weapon for the further development
-        and transformation of the thing. Doing this for "set theory" requires
-        taking account of the experience that the main pairs of opposing
-        tendencies in mathematics take the form of adjoint functors, and frees
-        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
-        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
-        metaphysical "construction". Further, experience with sheaves,
-        permutation representations, algebraic spaces, etc., shows that a "set
-        theory" for geometry should apply not only to abstract sets divorced
-        from time, space, ring of definition, etc., but also to more general
-        sets which do in fact develop along such parameters. For such sets,
-        usually logic is "intuitionistic" (in its formal properties)
-        usually the axiom of choice is false, and usually a set is not
-        determined by its points defined over <fr:tex>1</fr:tex> only.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    I have not read this, and have no idea what "contradictions" he's referring to.
-    <fr:link
-href="lawvere-1970.pdf"
-type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I have not read this. This might be a good source, aside from <fr:link
-href="ronnie-chen.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="ronnie-chen">Ronnie Chen</fr:link>,
-    where the viewpoint of "generalized algebraic semantics" is developed.
-</fr:p><fr:p><fr:link
-href="marquis-reyes-2012.pdf"
-type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Some basic questions, whose "elevator pitch" answers ought to roll off
         the tongue:
         <fr:ul><fr:li>
@@ -2730,9 +3161,10 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
                 Is it the same thing as "topos logic"?
             </fr:li></fr:ul>
         Following <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
         Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
@@ -2759,37 +3191,41 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sket
         complete the sentence".
         A truly FAQ-style answer compendium will have to come later-- after I've
         actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -2803,65 +3239,75 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -2877,107 +3323,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -2992,148 +3461,196 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -3154,248 +3671,377 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -3403,7 +4049,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -3411,156 +4058,544 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The [language] is sorted…. We inductively define terms and their sorts:
+            <fr:ul><fr:li>
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
+                </fr:li>
+                <fr:li>
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
+                </fr:li></fr:ul>"
+            <fr:ul><fr:li>
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
+            these terms in any elementary topos, developed in 13.3 and 13.4],
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
+        </fr:p><fr:p>
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Ignoring the natural numbers type and the idiosyncratic list type of
+        <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+        elementary topops for reference) of <fr:link
+type="local"
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
+        predicate logic" aka "topos logic" aka "higher-order intuitionistic
+        logic"</fr:link>, although certain inter-definable choices can be
+        made.
+    </fr:p><fr:p>
+        There's no clear notion of "taking everything", but a reasonable
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
+    </fr:p><fr:p>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
+<fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex
+display="inline"><![CDATA[∈]]></fr:tex>) left behind by the
+        process of accumulating (<fr:tex
+display="inline"><![CDATA[∪]]></fr:tex>) the power set (<fr:tex
+display="inline"><![CDATA[\mathcal {P}]]></fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> only.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+type="external"
+href="lawvere-1970.pdf">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+type="local"
+href="ronnie-chen.xml"
+addr="ronnie-chen"
+title="Ronnie Chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+type="external"
+href="marquis-reyes-2012.pdf">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+type="local"
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">Equality Between Functionals</fr:link> and <fr:link
+type="local"
+href="statman-1985.xml"
+addr="statman-1985"
+title="Equality Between Functionals Revisited">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+type="external"
+href="https://proofassistants.stackexchange.com/a/1177">stack overflow</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "The [language] is sorted…. We inductively define terms and their sorts:
-            <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
-                </fr:li>
-                <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
-                </fr:li></fr:ul>"
-            <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex
+display="inline"><![CDATA[⊢ s = t]]></fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+type="external"
+href="friedman-1975.pdf">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
+type="local"
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="statman-1985.pdf">statman-1985.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Fibered Right Adjoints are automatically fibered?
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
-            these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
-        </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Ignoring the natural numbers type and the idiosyncratic list type of
-        <fr:link
-href="johnstone-2002.xml"
-type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
-        elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
-        predicate logic" aka "topos logic" aka "higher-order intuitionistic
-        logic"</fr:link>, although certain inter-definable choices can be
-        made.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex
+display="inline"><![CDATA[∀M]]></fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
     </fr:p><fr:p>
-        There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
-    </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
-<fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     While reading <fr:link
-href="von-neumann-1925.xml"
 type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
     "that's… the point"-- "realization" that first order logic is the
     meta-language of axiomatic mathematics, just as (it seems) logical
     frameworks are the meta-languages of (typically λ-calculus based)
@@ -3569,25 +4604,28 @@ addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had th
 </fr:p><fr:p>
     That is, (classical) FoL is often the setting in which axiomatic
     mathematical theories are formalized. Classic (no pun intended) examples
-    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
+    are <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{group} = \{e, ⋅, -⁻¹\}, Th(\textsf {groups}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{DLO} = \{≤\}, Th(\textsf {Dense Linear Orders}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{sets} = \{∈\}, ZFC)]]></fr:tex>.
     (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
 </fr:p><fr:p>
     "definite property"
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -3601,14 +4639,14 @@ root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:ta
         computability</fr:strong>, logical relations are manipulated as if they were sets,
         making the essence of many complex logical relations arguments
         accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -3621,150 +4659,60 @@ root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</f
         and I just don't have some context, or it's actually presented almost
         mystically.
         The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+href="carette-kiselyov-shan-2007.xml"
+addr="carette-kiselyov-shan-2007"
+title="Finally Tagless, Partially Evaluated">finally tagless encodings</fr:link>
         and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
+type="external"
+href="hoffman-1999">a justification for hoas</fr:link> doesn't seem super
         on-the-point, but if it's mentioned it must be for good reason…
     </fr:p><fr:p>
         Again, quotes are air-quotes, not actual quotes.
         <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="sterling-2022.pdf">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
     <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>428</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
-    </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
-        <fr:ul><fr:li>
-                completness theorems,
-            </fr:li>
-            <fr:li>
-                the solvability of higher type functional equations, and
-            </fr:li>
-            <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="blass-2003.pdf">blass-2003.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Fibered Right Adjoints are automatically fibered?
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     What is the actual <fr:em>logic</fr:em> when doing a logical relation?
     Do you only actually need functors? Zeilberger's <fr:link
-href="millies-zeilberger-2015.pdf"
-type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
+type="external"
+href="millies-zeilberger-2015.pdf">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
     Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
 </fr:p><fr:p>
     Talked to Max about this on 5-9.
@@ -3772,86 +4720,71 @@ type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>m
     a hoare logic, and a implicational (eg FO) logic.
     I suppose one'd want to study a bit the relationship between the two.
     <fr:link
-href="millies-zeilberger-2015.xml"
 type="local"
-addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+href="millies-zeilberger-2015.xml"
+addr="millies-zeilberger-2015"
+title="Functors are Type Refinement Systems">paper</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>551</fr:anchor><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:taxon>Person</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>439</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Short About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    I am a master's student in Computer Science and Engineering at the
+    University of Michigan.
+    I'm interested in logic and related areas of computer science, particularly
+    programming languages.
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    This is my <fr:link
+type="external"
+href="https://www.jonmsterling.com/jms-005P.xml">forest</fr:link>.
+</fr:p><fr:ul><fr:li>
+                email: <fr:embedded-tex
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble><![CDATA[]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[hejohns@umich.edu]]></fr:embedded-tex-body></fr:embedded-tex></fr:li>
+    <fr:li>
+        photo:
+        <fr:p><html:img
+xmlns:html="http://www.w3.org/1999/xhtml"
+src="hejohns.jpeg"
+width="150"
+height="150" /></fr:p></fr:li>
+    <fr:li>
+        cv: <fr:link
+type="external"
+href="cv.pdf">cv.pdf</fr:link> (TODO)
+    </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree>
+<fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#205</fr:addr><fr:route>unstable-205.xml</fr:route><fr:title>Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
-</fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        My screen name is <fr:code>hejohns</fr:code>, eg <fr:link
+type="external"
+href="https://github.com/hejohns">https://github.com/hejohns</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000P.xml b/_site/forest/hejohns-000P.xml
index 8e9ad1e..c3896cf 100644
--- a/_site/forest/hejohns-000P.xml
+++ b/_site/forest/hejohns-000P.xml
@@ -1,13 +1,14 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>482</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p /></fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000Q.xml b/_site/forest/hejohns-000Q.xml
index d3a9be3..2ac6a09 100644
--- a/_site/forest/hejohns-000Q.xml
+++ b/_site/forest/hejohns-000Q.xml
@@ -1,14 +1,14 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html"
-type="external">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
-href="stack-overflow/every-topos-is-a-regular-category.html"
-type="external">stack-overflow/every-topos-is-a-regular-category.html</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>490</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
+type="external"
+href="stack-overflow/every-topos-is-a-regular-category.html">stack-overflow/every-topos-is-a-regular-category.html</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns-000R.xml b/_site/forest/hejohns-000R.xml
index d893a20..ed2faa2 100644
--- a/_site/forest/hejohns-000R.xml
+++ b/_site/forest/hejohns-000R.xml
@@ -1,13 +1,14 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>hejohns-000R</fr:addr><fr:route>hejohns-000R.xml</fr:route><fr:title>BLANK</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>17</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>hejohns-000R</fr:addr><fr:route>hejohns-000R.xml</fr:route><fr:title>BLANK</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>17</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hejohns.xml b/_site/forest/hejohns.xml
index f95d021..9b41640 100644
--- a/_site/forest/hejohns.xml
+++ b/_site/forest/hejohns.xml
@@ -1,259 +1,324 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>799</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:taxon>Person</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>271</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Short About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Short About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     I am a master's student in Computer Science and Engineering at the
     University of Michigan.
     I'm interested in logic and related areas of computer science, particularly
     programming languages.
 </fr:p><fr:p>
     This is my <fr:link
-href="https://www.jonmsterling.com/jms-005P.xml"
-type="external">forest</fr:link>.
+type="external"
+href="https://www.jonmsterling.com/jms-005P.xml">forest</fr:link>.
 </fr:p><fr:ul><fr:li>
                 email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li>
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble><![CDATA[]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[hejohns@umich.edu]]></fr:embedded-tex-body></fr:embedded-tex></fr:li>
     <fr:li>
         photo:
         <fr:p><html:img
 xmlns:html="http://www.w3.org/1999/xhtml"
 src="hejohns.jpeg"
 width="150"
-height="150"></html:img></fr:p></fr:li>
+height="150" /></fr:p></fr:li>
     <fr:li>
         cv: <fr:link
-href="cv.pdf"
-type="external">cv.pdf</fr:link> (TODO)
-    </fr:li></fr:ul></fr:mainmatter></fr:tree>
+type="external"
+href="cv.pdf">cv.pdf</fr:link> (TODO)
+    </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>273</fr:anchor><fr:title>Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#205</fr:addr><fr:route>unstable-205.xml</fr:route><fr:title>Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         My screen name is <fr:code>hejohns</fr:code>, eg <fr:link
-href="https://github.com/hejohns"
-type="external">https://github.com/hejohns</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+type="external"
+href="https://github.com/hejohns">https://github.com/hejohns</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>727</fr:anchor><fr:addr>hejohns-000R</fr:addr><fr:route>hejohns-000R.xml</fr:route><fr:title>BLANK</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>17</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>729</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="true"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:meta
+name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>731</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Basic information</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>219</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     I am a master's student in Computer Science and Engineering at the
     University of Michigan.
     I'm interested in logic and related areas of computer science, particularly
     programming languages.
 </fr:p><fr:p>
     This is my <fr:link
-href="https://www.jonmsterling.com/jms-005P.xml"
-type="external">forest</fr:link>.
+type="external"
+href="https://www.jonmsterling.com/jms-005P.xml">forest</fr:link>.
 </fr:p><fr:ul><fr:li>
                 email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li>
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble><![CDATA[]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[hejohns@umich.edu]]></fr:embedded-tex-body></fr:embedded-tex></fr:li>
     <fr:li>
         photo:
         <fr:p><html:img
 xmlns:html="http://www.w3.org/1999/xhtml"
 src="hejohns.jpeg"
 width="150"
-height="150"></html:img></fr:p></fr:li>
+height="150" /></fr:p></fr:li>
     <fr:li>
         cv: <fr:link
-href="cv.pdf"
-type="external">cv.pdf</fr:link> (TODO)
-    </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="cv.pdf">cv.pdf</fr:link> (TODO)
+    </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree>
+    <fr:p>
+        For more information about me, see <fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link>.
+    </fr:p>
+<fr:tree
 toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>733</fr:anchor><fr:addr>hejohns-000N</fr:addr><fr:route>hejohns-000N.xml</fr:route><fr:title>L (Language)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>223</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    I vaguely recall reading that the tradition of isolating the non-logical
-    symbols (constants, functions, relations) of a first order logic for a
-    language <fr:tex>\mathcal {L}</fr:tex>, and referring to suitable models as
-    <fr:tex>\mathcal {L}</fr:tex>-structures, was popularized by Carnap, and maybe in
-    particular by <fr:em>The Logical Syntax of Language</fr:em>.
-</fr:p><fr:p>
-    I am unable to track down a source for this now. I'll have to check <fr:em>The
-    Syntax</fr:em> sometime. I can say at least <fr:em>Meaning and Necessity</fr:em> does
-    <fr:em>not</fr:em> make use of this terminology in the same way.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>735</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Many authors object to the name "higher-order intuitionistic logic".
-</fr:p><fr:p><fr:link
-href="blass-1988.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic." (p. 2).
-</fr:p><fr:p>
-    And <fr:link
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
-    axioms are used to define the higher order internal language. The logic is
-    called 'topos logic' here, although 'intuitionistic logic' is often
-    referred to in the literature. I believe that 'intuitionism' is usually,
-    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
-    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
-    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
-    (1979). But topos logic coincides with no intuitionistic logic studied
-    before toposes, and this is due to real philosophical differences (see
-    McLarty 1990a)." (p. vii).
-</fr:p><fr:p>
-    Of course this discrepancy is what one should come to expect, since the
-    very notion of a formalized logic of intuitionistic reasoning is at odds
-    with Brouwer's notion of intuitionism, so when one speaks of
-    "intuitionistic logic" it's usually in the purely formal sense.
-</fr:p><fr:p>
-    Combining the two names, I think "higher-order intuitionistic topos logic"
-    is at least a little more descriptive.
-    Although I like calling it "finite-type intuitionistic predicate logic" to
-    keep in touch with some older terminology you still see around the block.
-
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>737</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
-            <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
-                </fr:li>
-                <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
-                </fr:li>
-                <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+type="local"
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+type="local"
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>739</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -269,107 +334,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -384,869 +472,587 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
-type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>741</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
-    (Recall that categories can be formalized as a two or dependently-sorted
-    first order theory.)
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
 </fr:p><fr:p>
-    This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
+    And <fr:link
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
-    that followed, Lawvere tried to extend his analysis and sketched a
-    categorical version of first-order theories under the name of elementary
-    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
-    introduced the notion of an elementary topos, making an explicit connection
-    with higher-order logic and type theories. Both Lawvere and Tierney were
-    aiming at an elementary, that is first-order, axiomatic presentation of
-    what are now called Grothendieck toposes, a special type of categories
-    introduced by Alexandre Grothendieck in the context of algebraic geometry
-    and sheaf theory. Soon after, connections with intuitionistic analysis,
-    recursive functions, completeness theorems for various logical systems,
-    differential geometry, constructive mathematics were made." (p. 2).
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
 </fr:p><fr:p>
-    If <fr:link
-href="lawvere-1970.xml"
-type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
-    it's curious to note that the word "elementary" never appears.
-    It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>743</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Some basic questions, whose "elevator pitch" answers ought to roll off
-        the tongue:
-        <fr:ul><fr:li>
-                What is a(n elementary) topos?
-            </fr:li>
-            <fr:li>
-                What is this "topos logic" that is sound in any elementary topos?
-                As in literally, what is the syntax?
-                Is it the same thing as this "higher-order intuitionistic logic"?
-                What are the inference rules/axioms?
-            </fr:li>
-            <fr:li>
-                What is this "Kripke-Joyal semantics"?
-                Is this the same thing as the so-called "forcing semantics"?
-            </fr:li>
-            <fr:li>
-                What is this "Mitchell-Bénabou language"?
-                Is it the same thing as "topos logic"?
-            </fr:li></fr:ul>
-        Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-    </fr:p><fr:p>
-        Some more difficult questions we do not attempt to answer at the moment:
-        <fr:ul><fr:li>
-                What is a Grothendieck topos, and what's the difference v.
-                elementary toposes?
-            </fr:li>
-            <fr:li>
-                What is a "geometric logic" and a "geometric morphism"?
-                And what's the difference v. a "logical morphism"?
-            </fr:li>
-            <fr:li>
-                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-                What is forcing??
-                Why does the concept of "generic" show up everywhere?
-            </fr:li>
-            <fr:li>
-                What's all this algebraic geometry stuff?
-            </fr:li>
-            <fr:li>
-                And everything else…
-            </fr:li></fr:ul></fr:p><fr:p>
-        Although as you might expect, we quickly derail from "elevator pitch"
-        answers, to "quick, write that down before I lose motivation and never
-        complete the sentence".
-        A truly FAQ-style answer compendium will have to come later-- after I've
-        actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+    Although I like calling it "finite-type intuitionistic predicate logic" to
+    keep in touch with some older terminology you still see around the block.
+
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
-    (Recall that categories can be formalized as a two or dependently-sorted
-    first order theory.)
-</fr:p><fr:p>
-    This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    There is some (a posteriori insignificant) variation in the presentation of
+    the syntax, thus following through with some input from <fr:link
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
-    that followed, Lawvere tried to extend his analysis and sketched a
-    categorical version of first-order theories under the name of elementary
-    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
-    introduced the notion of an elementary topos, making an explicit connection
-    with higher-order logic and type theories. Both Lawvere and Tierney were
-    aiming at an elementary, that is first-order, axiomatic presentation of
-    what are now called Grothendieck toposes, a special type of categories
-    introduced by Alexandre Grothendieck in the context of algebraic geometry
-    and sheaf theory. Soon after, connections with intuitionistic analysis,
-    recursive functions, completeness theorems for various logical systems,
-    differential geometry, constructive mathematics were made." (p. 2).
-</fr:p><fr:p>
-    If <fr:link
-href="lawvere-1970.xml"
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
+    juxtapose multiple definitions-- <fr:link
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
-    it's curious to note that the word "elementary" never appears.
-    It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
+    topoi</fr:link>--, but this time we slightly modify the source for
+    uniformity as needed.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
-    Of course, these are always the kind of descriptions that only make sense
-    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
-    We remark that <fr:link
-href="mclarty-1990b.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
-    perspective.
-</fr:p><fr:p>
-    Instead, I've gathered every book I could conveniently get my hands on just
-    now, and we'll see how each defines an elementary topos.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
+        <fr:ul><fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
+            <fr:li>
+                a class of <fr:em>terms</fr:em> of each type, including countably many
+                variables of each type
+            </fr:li>
+            <fr:li>
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
+            </fr:li></fr:ul>"
+    </fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="lambek-scott-1986.xml"
-type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
-        classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The class of types is closed under the inductive clauses:
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="johnstone-2002.xml"
-type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "We are now ready to present the definition of an (elementary) topos,
-        which is the central notion of this book: it is a category having all
-        the good 'set-like' properties discussed in the previous chapter,
-        except (possibly) for Booleanness. It is a remarkable fact that all
-        these properties are implied by the conjunction of just two of them:"
-    </fr:p><fr:p>
-        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
-        classifier."
-    </fr:p><fr:p>
-        "… Before that, however, we must mention that the above definition of a
-        topos, though it is appreciably simplified from that originally adopted
-        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
-        not the simplest possible; one can combine the exponentials and the
-        subobject classifier into a single structure, that of power objects."
-    </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
-        products and all its objects are exponentiable. … Nevertheless, most of
-        the cartesian closed categories we shall meet will have equalizers as
-        well as finite products; we shall indicate this (with a note of mild
-        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The class of terms is freely generated from certain basic terms by
+            certain operations including the following… and is closed under certain operations.
+            <fr:ul><fr:li>
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
+                </fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
-        classifier."
-    </fr:p><fr:p>
-        Note though, that McLarty "assumes a base category with finite limits"
-        when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
+        </fr:p><fr:p>
+            "One may also define a number of widely used term forming
+            operations: …
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="makkai-reyes-1977.xml"
-type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Okay so they only define a Grothendieck topos. We'll leave the
-        discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (We write "language" whenever Johnstone writes "signature", and use the
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
+        I don't know if Johnstone prefers the term "signature" because
+        "language" is overloaded and often refers not just to the, well,
+        language, but to the larger logic sans deductive system, but that would
+        be a good reason.)
+    </fr:p><fr:p>
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        of <fr:em>type constructors</fr:em> chosen from the following list, which
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
+    </fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="awodey-2010.xml"
-type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
-        <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+                (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
+                type.
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
-            </fr:li></fr:ul>
-        This compact definition proves to be amazingly rich in consequences:
-        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
+                a type.
+            </fr:li>
+            <fr:li>
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
+            </fr:li>
+            <fr:li>
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
+            </fr:li></fr:ul><fr:p>
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
+        </fr:p><fr:p>
+            "If our language has the type constructors (ii) and (iv), we shall
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
+        </fr:p><fr:p>
+            "From now on, all our languages will contain the product type
+            constructor; thus we have restricted all our primitive function and
+            relation symbols to be unary. (If our signature also contaisn
+            function types, then we can further restrict our function symbols
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
+            primitive relation symbols by constants as well)"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+        "The term constructor for power types in fact builds terms form
+        formulae, so over a language which contains power types we have to
+        define terms and formulae by a simultaneous recursion".
+    </fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="goldblatt-2006.xml"
-type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
-        <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
-            </fr:li>
-            <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
-            </fr:li>
-            <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
-            </fr:li>
-            <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
-            </fr:li></fr:ul>
-        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
-        <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
-            </fr:li></fr:ul>
-        and dually (2) replaced by
-        <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
-            </fr:li></fr:ul>
-        The definition just given is the one originally proposed by Lawvere and
-        Tierney, in terms of which they started [elementary] topos theory in
-        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
-        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
-        a topos can be defined as a Cartesian closed category with a subobject
-        classifier. In 4.7 we shall consider a different definition, based on a
-        categorial characterisation of power sets."
-    </fr:p><fr:p>
-        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
-        complete</fr:em>, meaning every finite diagram (finite objects, finite
-        morphisms) has a limit, "constitute", meaning that "cartesian closed"
-        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
-        <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
-type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
-type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
-            <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
-                </fr:li>
-                <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
-                </fr:li>
-                <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
-type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
-            <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
-            classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
-            any category which satisfies (1) and has all exponentials will, in
-            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
-type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
-            course of this chapter that the two sets of axioms are actually
-            equivalent.
-        </fr:p><fr:p>
-            Notice that the two natural isomorphisms (1) and (2) may be combined
-            in a single isomorphism
-            <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
-            also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
-    every definition, and give a standard one:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
-    <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
-    is a pullback square.
-</fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
-type="local"
-addr="hejohns-000K">global
-    element</fr:link> is trivially monic. Recall that "monos are stable
-    under pullback".
-</fr:p><fr:p>
-    This is from <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
-    So comparing each definition with another, they're all about par for
-    course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Many authors object to the name "higher-order intuitionistic logic".
-</fr:p><fr:p><fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic." (p. 2).
-</fr:p><fr:p>
-    And <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
-    axioms are used to define the higher order internal language. The logic is
-    called 'topos logic' here, although 'intuitionistic logic' is often
-    referred to in the literature. I believe that 'intuitionism' is usually,
-    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
-    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
-    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
-    (1979). But topos logic coincides with no intuitionistic logic studied
-    before toposes, and this is due to real philosophical differences (see
-    McLarty 1990a)." (p. vii).
-</fr:p><fr:p>
-    Of course this discrepancy is what one should come to expect, since the
-    very notion of a formalized logic of intuitionistic reasoning is at odds
-    with Brouwer's notion of intuitionism, so when one speaks of
-    "intuitionistic logic" it's usually in the purely formal sense.
-</fr:p><fr:p>
-    Combining the two names, I think "higher-order intuitionistic topos logic"
-    is at least a little more descriptive.
-    Although I like calling it "finite-type intuitionistic predicate logic" to
-    keep in touch with some older terminology you still see around the block.
-
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    There is some (a posteriori insignificant) variation in the presentation of
-    the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
-type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
-    juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
-type="local"
-addr="hejohns-000G">as done for elementary
-    topoi</fr:link>--, but this time we slightly modify the source for
-    uniformity as needed.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
-type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
-        <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
-            <fr:li>
-                a class of <fr:em>terms</fr:em> of each type, including countably many
-                variables of each type
-            </fr:li>
-            <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
-            </fr:li></fr:ul>"
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "The class of terms is freely generated from certain basic terms by
-            certain operations including the following… and is closed under certain operations.
-            <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
-                </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
-                </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
-                </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
-                </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
-        </fr:p><fr:p>
-            "One may also define a number of widely used term forming
-            operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
-                </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
-type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
-        I don't know if Johnstone prefers the term "signature" because
-        "language" is overloaded and often refers not just to the, well,
-        language, but to the larger logic sans deductive system, but that would
-        be a good reason.)
-    </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
-        of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-                (i) <fr:em>Basic types</fr:em>: Each sort is a type
-            </fr:li>
-            <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
-                type.
-            </fr:li>
-            <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
-                a type.
-            </fr:li>
-            <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
-            </fr:li>
-            <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
-            </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
-        </fr:p><fr:p>
-            "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
-        </fr:p><fr:p>
-            "From now on, all our languages will contain the product type
-            constructor; thus we have restricted all our primitive function and
-            relation symbols to be unary. (If our signature also contaisn
-            function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
-            primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
-        "The term constructor for power types in fact builds terms form
-        formulae, so over a language which contains power types we have to
-        define terms and formulae by a simultaneous recursion".
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -1254,7 +1060,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -1262,772 +1069,749 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>270</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>745</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    While reading <fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
-    "that's… the point"-- "realization" that first order logic is the
-    meta-language of axiomatic mathematics, just as (it seems) logical
-    frameworks are the meta-languages of (typically λ-calculus based)
-    programming languages. Well not literally, but the discrepancy is
-    non-substantial.
-</fr:p><fr:p>
-    That is, (classical) FoL is often the setting in which axiomatic
-    mathematical theories are formalized. Classic (no pun intended) examples
-    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
-    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
-</fr:p><fr:p>
-    "definite property"
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>747</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>749</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>751</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>753</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>755</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
+        Thus <fr:link
 type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
 type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
-    </fr:p><fr:p>
-        The <fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
+        </fr:li>
 
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
-            <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
-                </fr:li>
-                <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        <fr:li>
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
-type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>757</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>759</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>761</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>763</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -2035,2030 +1819,2975 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
+type="local"
 href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    However, unlike <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>765</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Fibered Right Adjoints are automatically fibered?
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>767</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
-    Do you only actually need functors? Zeilberger's <fr:link
-href="millies-zeilberger-2015.pdf"
-type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
-    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
-</fr:p><fr:p>
-    Talked to Max about this on 5-9.
-    They really just give to different notions of "logic over syntax"--
-    a hoare logic, and a implicational (eg FO) logic.
-    I suppose one'd want to study a bit the relationship between the two.
-    <fr:link
-href="millies-zeilberger-2015.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>769</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
-    </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
+        <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
         The <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
-        </fr:li>
-        <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
+    </fr:p><fr:p>
+        This proof is rather low tech and is lifted straight from <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
+        paper</fr:link>. It's also how you'd attack <fr:link
 type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
+        Then <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
-type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+type="local"
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+type="local"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
+                    <fr:embedded-tex
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="subb-0006.xml"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
+    
+    <fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
+    course text(s). I haven't used it much for reading either, owing to the
+    large amount of new notation one has to refresh every time you open the
+    book, but it's a solid reference book.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
     </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex
+display="inline"><![CDATA[∈]]></fr:tex>) left behind by the
+        process of accumulating (<fr:tex
+display="inline"><![CDATA[∪]]></fr:tex>) the power set (<fr:tex
+display="inline"><![CDATA[\mathcal {P}]]></fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> only.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+type="external"
+href="lawvere-1970.pdf">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The view that toposes originated as generalized set theory is a figment of
+        set theoretically educated common sense. This false history obstructs
+        understanding of category theory and especially of categorical foundations
+        for mathematics. Problems in geometry, topology, and related algebra led to
+        categories and toposes. Elementary toposes arose when Lawvere's interest in
+        the foundations of physics and Tierney's in the foundations of topology led
+        both to study Grothendieck's foundations for algebraic geometry. I end with
+        remarks on a categorical view of the history of set theory, including a
+        false history plausible from that point of view that would make it helpful
+        to introduce toposes as a generalization from set theory.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="mclarty-1990b.pdf">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
+href="ronnie-chen.xml"
+addr="ronnie-chen"
+title="Ronnie Chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+type="external"
+href="marquis-reyes-2012.pdf">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">Equality Between Functionals</fr:link> and <fr:link
+type="local"
+href="statman-1985.xml"
+addr="statman-1985"
+title="Equality Between Functionals Revisited">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+type="external"
+href="https://proofassistants.stackexchange.com/a/1177">stack overflow</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex
+display="inline"><![CDATA[⊢ s = t]]></fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+type="external"
+href="friedman-1975.pdf">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="statman-1985.pdf">statman-1985.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We have built the first family of tagless interpretations for a
+        higher-order typed object language in a typed metalanguage (Haskell or
+        ML) that require no dependent types, generalized algebraic data types,
+        or postprocessing to eliminate tags. The statically type-preserving
+        interpretations include an evaluator, a compiler (or staged evaluator),
+        a partial evaluator, and call-by-name and call-by-value CPS
+        transformers.
+    </fr:p><fr:p>
+        Our main idea is to encode HOAS using cogen functions rather than data
+        constructors. In other words, we represent object terms not in an
+        initial algebra but using the coalgebraic structure of the λ-calculus.
+        Our representation also simulates inductive maps from types to types,
+        which are required for typed partial evaluation and CPS
+        transformations.
+    </fr:p><fr:p>
+        Our encoding of an object term abstracts over the various ways to
+        interpret it, yet statically assures that the interpreters never get
+        stuck.  To achieve self-interpretation and show Jones-optimality, we
+        relate this exemplar of higher-rank and higher-kind polymorphism to
+        plugging a term into a context of let-polymorphic bindings.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the APLAS version (conference paper). There is a roughly twice
+        as long journal version.
+    </fr:p><fr:p>
+        I didn't study a lot of the probably more interesting aspects of this
+        work.
+        I just wanted a first/resonant-understanding of "what does <fr:em>finally
+        tagless</fr:em> mean?".
+    </fr:p><fr:p>
+        It seems the very basic idea is about the general problem of DSLs.
+        The very short way to say it, is that we like to construct languages as
+        object languages of nice meta languages.
+        For example, a nice meta languages gives us nice primitives to reuse as
+        the denotational semantics of the object language.
+        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
+        the semantics of the DSL/embedded/object language's numeric type.
+    </fr:p><fr:p>
+        Most people's knee jerk reaction is to encode the object language as an
+        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
+        that it is the free syntax generated by the encoded rules/initial
+        algebra/inductive dataype.
+        But the naïve TODO
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex
+display="inline"><![CDATA[∀M]]></fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
     </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="carette-kiselyov-shan-2007.xml"
+addr="carette-kiselyov-shan-2007"
+title="Finally Tagless, Partially Evaluated">finally tagless encodings</fr:link>
+        and <fr:link
+type="external"
+href="hoffman-1999">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+type="external"
+href="sterling-2022.pdf">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+type="external"
+href="blass-2003.pdf">blass-2003.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="blass-1988.pdf">blass-1988.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    One of the early NBG papers. Maybe "the" NBG paper.
+    My notes on this are currently in <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
+href="quine-1960.xml"
+addr="quine-1960"
+title="Variables Explained Away">Variables Explained Away</fr:link>.
+</fr:p><fr:p>
+    English translation in <fr:link
 type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+href="van-heijenoort-1967.xml"
+addr="van-heijenoort-1967"
+title="From Frege To Gödel">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A+. I should get a second copy to gift whenever needed.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
+    this list. Whatever else you might think about Quine, he was a terrific
+    explainer, and this is the best introduction to the subject I know."
+</fr:p><fr:p>
+    (Yes, that 50 should be a 60(b).)
+    Here is the followup I wrote:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
     </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
+        I was surprised by the mention of von Neumann 1925 in this snippet:
     </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
     </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
-            <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
-                </fr:li>
-                <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+type="external"
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:ul><fr:li><fr:link
+type="external"
+href="quine-1960.pdf">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    The first point:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        During my Topology lectures at Macquarie University in the 1970s, the
+        students and I realized, in proving that a function f between posets was
+        order preserving when there was a function u in the reverse direction such
+        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
+        preserving. I realized then that knowing functors in the two directions
+        only on objects and the usual hom adjointness isomorphism implied the
+        effect of the functors on homs was uniquely determined. Writing this down
+        properly led to the present paper.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
-type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
+        sympathy to this line of thought, I’m skeptical of the unspoken assumption
+        that there’s an underlying concept of a ‘foundation’ up for analysis, that
+        this analysis would properly guide our assessment of the various
+        candidates. In contrast, it seems to me that the considerations the
+        combatants offer against opponents and for their preferred candidates, as
+        well as the roles each candidate actually or potentially succeeds in
+        playing, reveal quite a number of different jobs that mathematicians want
+        done. What matters is these jobs we want our theories to do and how well
+        they do them. Whether any or all of them, jobs or theories, deserves to be
+        called ‘foundational’ is really beside the point.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    A nice article that can be briefly summarized as
+    <fr:ul><fr:li>
+            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
+            system, whatever that means
+        </fr:li>
+        <fr:li>
+            Classical set theory is perfectly fine as a foundational system in
+            the traditionally (ie ZFC) conceived sense
+        </fr:li>
+        <fr:li>
+            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
+            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
+            bleeding edge").
+        </fr:li></fr:ul></fr:p><fr:p>
+    The contrast between the minimalistic ZFC and potentially enormous "user
+    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
+    natural deduction. Indeed, it's like how Ronnie calls set theory "the
+    machine/assembly language of math" (kind of paraphrase), and Max calls SK
+    combinators "the machine/assembly language of λ-calculus".
+</fr:p><fr:p>
+    Natural deduction is certainly more complicated than SK/Hilbert proof
+    systems, but has nice usability properties that are worth study and
+    implementation.
+    The general mathematical trend seems to be that of starting with a spartan
+    formal system, then gradually making it more user friendly to accomodate
+    every increasingly complicated mathematics, where the minimalistic system
+    starts to leave one wanting.
+</fr:p><fr:p>
+    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
+    redundant to mention other proof assistants eg Cubical Agda.
+</fr:p><fr:p>
+    I've used quotes here as air quotes, not actual quotes of Maddy.
+    <fr:link
+type="external"
+href="maddy-2019.pdf">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>771</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>773</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+type="external"
+href="zeilberger">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>775</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
+<fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>777</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Contributing to <fr:link
+type="external"
+href="https://github.com/maxsnew/cubical-categorical-logic">Cubical Categorical Logic</fr:link>, "cubical agda formalizations of different approaches to categorical semantics of type theories"
+        </fr:li>
+        <fr:li><fr:link
+type="external"
+href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>779</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+type="external"
+href="https://mplse-reading-group.github.io/">https://mplse-reading-group.github.io/</fr:link></fr:li>
+        <fr:li><fr:link
+type="external"
+href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
+type="external"
+href="../posts.html">Jekyll Blog</fr:link></fr:mainmatter><fr:backmatter /></fr:tree>
+    
+    <fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p /></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
+type="external"
+href="stack-overflow/every-topos-is-a-regular-category.html">stack-overflow/every-topos-is-a-regular-category.html</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+type="external"
+href="https://github.com/tomdjong/MGS-categorical-realizability">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
         </fr:li>
-
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
-        </fr:li>
-        <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
 type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>781</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
+        classifier and a natural numbers object."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
     </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
-type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
+href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>783</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
-href="hejohns-0007.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>785</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>787</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>271</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Short About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    I am a master's student in Computer Science and Engineering at the
-    University of Michigan.
-    I'm interested in logic and related areas of computer science, particularly
-    programming languages.
-</fr:p><fr:p>
-    This is my <fr:link
-href="https://www.jonmsterling.com/jms-005P.xml"
-type="external">forest</fr:link>.
-</fr:p><fr:ul><fr:li>
-                email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li>
-    <fr:li>
-        photo:
-        <fr:p><html:img
-xmlns:html="http://www.w3.org/1999/xhtml"
-src="hejohns.jpeg"
-width="150"
-height="150"></html:img></fr:p></fr:li>
-    <fr:li>
-        cv: <fr:link
-href="cv.pdf"
-type="external">cv.pdf</fr:link> (TODO)
-    </fr:li></fr:ul></fr:mainmatter></fr:tree>
-<fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+type="local"
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>273</fr:anchor><fr:title>Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        My screen name is <fr:code>hejohns</fr:code>, eg <fr:link
-href="https://github.com/hejohns"
-type="external">https://github.com/hejohns</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course, if we ignore natural numbers objects for now.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>789</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+    Although I like calling it "finite-type intuitionistic predicate logic" to
+    keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>791</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    There is some (a posteriori insignificant) variation in the presentation of
+    the syntax, thus following through with some input from <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
+    juxtapose multiple definitions-- <fr:link
+type="local"
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
+    topoi</fr:link>--, but this time we slightly modify the source for
+    uniformity as needed.
+</fr:p><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>793</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
+        <fr:ul><fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
+            <fr:li>
+                a class of <fr:em>terms</fr:em> of each type, including countably many
+                variables of each type
+            </fr:li>
+            <fr:li>
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
+            </fr:li></fr:ul>"
+    </fr:p><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>795</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The class of types is closed under the inductive clauses:
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The class of terms is freely generated from certain basic terms by
+            certain operations including the following… and is closed under certain operations.
+            <fr:ul><fr:li>
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
+                </fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
+        </fr:p><fr:p>
+            "One may also define a number of widely used term forming
+            operations: …
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (We write "language" whenever Johnstone writes "signature", and use the
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
+        I don't know if Johnstone prefers the term "signature" because
+        "language" is overloaded and often refers not just to the, well,
+        language, but to the larger logic sans deductive system, but that would
+        be a good reason.)
+    </fr:p><fr:p>
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        of <fr:em>type constructors</fr:em> chosen from the following list, which
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
+    </fr:p><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>797</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+                (i) <fr:em>Basic types</fr:em>: Each sort is a type
+            </fr:li>
+            <fr:li>
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
+                type.
+            </fr:li>
+            <fr:li>
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
+                a type.
+            </fr:li>
+            <fr:li>
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
+            </fr:li>
+            <fr:li>
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
+            </fr:li></fr:ul><fr:p>
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
+        </fr:p><fr:p>
+            "If our language has the type constructors (ii) and (iv), we shall
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
+        </fr:p><fr:p>
+            "From now on, all our languages will contain the product type
+            constructor; thus we have restricted all our primitive function and
+            relation symbols to be unary. (If our signature also contaisn
+            function types, then we can further restrict our function symbols
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
+            primitive relation symbols by constants as well)"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+        "The term constructor for power types in fact builds terms form
+        formulae, so over a language which contains power types we have to
+        define terms and formulae by a simultaneous recursion".
+    </fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>439</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            (We omit the bookkeeping of free variables that Johnstone inlines with
+            the following definitions.)
+            "<fr:ul><fr:li>
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (iii)
+                    <fr:ul><fr:li>
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
+                        </fr:li>
+                        <fr:li>
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
+                        </fr:li></fr:ul></fr:li>
+                <fr:li>
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
+                    [This is formally different from (ii).]
+                </fr:li>
+                <fr:li>
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (viii)
+                    <fr:ul><fr:li>
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
+                        </fr:li>
+                        <fr:li>
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
+                        </fr:li></fr:ul></fr:li>
+                <fr:li>
+                    (ix) The remaining term constructor associated with list
+                    types, the <fr:em>iterator</fr:em>, is more complicated [which we
+                    omit for irrelevance].
+                </fr:li></fr:ul></fr:p><fr:p>
+            Of course, clauses (v) and (vi) of the definition above apply only
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
+            types, and clauses (viii) and (ix) only if we have list types.
+            Note that, except for power types, the clauses come in pairs [of
+            constructors and destructors]. The reason why we have no
+            'destructor' for power types is that, just as the constructor for
+            power types builds terms out of formulae rather than terms, the
+            corresponding destructor builds formulae out of terms of a power
+            type-- it thus appears as clause (iii) of the next definition."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
-</fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            (We again omit the bookkeeping of free variables.)
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
+            <fr:ul><fr:li>
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
+                </fr:li>
+                <fr:li>
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
+                </fr:li>
+                <fr:li>
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
+                </fr:li></fr:ul></fr:p><fr:p>
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
+        interpretation), instead of describing the logic abstractly and taking
+        the internal language to be a diagram (in Tarskian-model theory speak)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+    </fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            The types have the structure of the objects of an elementary topos,
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="false"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="true"><fr:frontmatter><fr:anchor>725</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:contributor><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors><fr:meta
-name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The [language] is sorted…. We inductively define terms and their sorts:
+            <fr:ul><fr:li>
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
+                </fr:li>
+                <fr:li>
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
+                </fr:li></fr:ul>"
+            <fr:ul><fr:li>
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>277</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
+            these terms in any elementary topos, developed in 13.3 and 13.4],
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
+        </fr:p><fr:p>
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Ignoring the natural numbers type and the idiosyncratic list type of
+        <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    I am a master's student in Computer Science and Engineering at the
-    University of Michigan.
-    I'm interested in logic and related areas of computer science, particularly
-    programming languages.
-</fr:p><fr:p>
-    This is my <fr:link
-href="https://www.jonmsterling.com/jms-005P.xml"
-type="external">forest</fr:link>.
-</fr:p><fr:ul><fr:li>
-                email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li>
-    <fr:li>
-        photo:
-        <fr:p><html:img
-xmlns:html="http://www.w3.org/1999/xhtml"
-src="hejohns.jpeg"
-width="150"
-height="150"></html:img></fr:p></fr:li>
-    <fr:li>
-        cv: <fr:link
-href="cv.pdf"
-type="external">cv.pdf</fr:link> (TODO)
-    </fr:li></fr:ul></fr:mainmatter></fr:tree>
-    <fr:p>
-        For more information about me, see <fr:link
-href="hejohns.xml"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+        elementary topops for reference) of <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link>.
-    </fr:p>
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
+        predicate logic" aka "topos logic" aka "higher-order intuitionistic
+        logic"</fr:link>, although certain inter-definable choices can be
+        made.
+    </fr:p><fr:p>
+        There's no clear notion of "taking everything", but a reasonable
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
+    </fr:p><fr:p>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
-numbered="true"
-show-heading="false"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>357</fr:anchor></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>311</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Some basic questions, whose "elevator pitch" answers ought to roll off
-        the tongue:
-        <fr:ul><fr:li>
-                What is a(n elementary) topos?
-            </fr:li>
-            <fr:li>
-                What is this "topos logic" that is sound in any elementary topos?
-                As in literally, what is the syntax?
-                Is it the same thing as this "higher-order intuitionistic logic"?
-                What are the inference rules/axioms?
-            </fr:li>
-            <fr:li>
-                What is this "Kripke-Joyal semantics"?
-                Is this the same thing as the so-called "forcing semantics"?
-            </fr:li>
-            <fr:li>
-                What is this "Mitchell-Bénabou language"?
-                Is it the same thing as "topos logic"?
-            </fr:li></fr:ul>
-        Following <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
     </fr:p><fr:p>
-        Some more difficult questions we do not attempt to answer at the moment:
-        <fr:ul><fr:li>
-                What is a Grothendieck topos, and what's the difference v.
-                elementary toposes?
-            </fr:li>
-            <fr:li>
-                What is a "geometric logic" and a "geometric morphism"?
-                And what's the difference v. a "logical morphism"?
-            </fr:li>
-            <fr:li>
-                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-                What is forcing??
-                Why does the concept of "generic" show up everywhere?
-            </fr:li>
-            <fr:li>
-                What's all this algebraic geometry stuff?
-            </fr:li>
-            <fr:li>
-                And everything else…
-            </fr:li></fr:ul></fr:p><fr:p>
-        Although as you might expect, we quickly derail from "elevator pitch"
-        answers, to "quick, write that down before I lose motivation and never
-        complete the sentence".
-        A truly FAQ-style answer compendium will have to come later-- after I've
-        actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex
+display="inline"><![CDATA[∈]]></fr:tex>) left behind by the
+        process of accumulating (<fr:tex
+display="inline"><![CDATA[∪]]></fr:tex>) the power set (<fr:tex
+display="inline"><![CDATA[\mathcal {P}]]></fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> only.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+type="external"
+href="lawvere-1970.pdf">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
+type="local"
+href="ronnie-chen.xml"
+addr="ronnie-chen"
+title="Ronnie Chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+type="external"
+href="marquis-reyes-2012.pdf">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+type="local"
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">Equality Between Functionals</fr:link> and <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="statman-1985.xml"
+addr="statman-1985"
+title="Equality Between Functionals Revisited">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+type="external"
+href="https://proofassistants.stackexchange.com/a/1177">stack overflow</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
-    (Recall that categories can be formalized as a two or dependently-sorted
-    first order theory.)
-</fr:p><fr:p>
-    This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
-type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
-    that followed, Lawvere tried to extend his analysis and sketched a
-    categorical version of first-order theories under the name of elementary
-    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
-    introduced the notion of an elementary topos, making an explicit connection
-    with higher-order logic and type theories. Both Lawvere and Tierney were
-    aiming at an elementary, that is first-order, axiomatic presentation of
-    what are now called Grothendieck toposes, a special type of categories
-    introduced by Alexandre Grothendieck in the context of algebraic geometry
-    and sheaf theory. Soon after, connections with intuitionistic analysis,
-    recursive functions, completeness theorems for various logical systems,
-    differential geometry, constructive mathematics were made." (p. 2).
-</fr:p><fr:p>
-    If <fr:link
-href="lawvere-1970.xml"
-type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
-    it's curious to note that the word "elementary" never appears.
-    It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex
+display="inline"><![CDATA[⊢ s = t]]></fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+type="external"
+href="friedman-1975.pdf">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
-    Of course, these are always the kind of descriptions that only make sense
-    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
-    We remark that <fr:link
-href="mclarty-1990b.xml"
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
+    </fr:p><fr:p>
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="statman-1985.pdf">statman-1985.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
-    perspective.
-</fr:p><fr:p>
-    Instead, I've gathered every book I could conveniently get my hands on just
-    now, and we'll see how each defines an elementary topos.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Fibered Right Adjoints are automatically fibered?
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="lambek-scott-1986.xml"
-type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
-        classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "We are now ready to present the definition of an (elementary) topos,
-        which is the central notion of this book: it is a category having all
-        the good 'set-like' properties discussed in the previous chapter,
-        except (possibly) for Booleanness. It is a remarkable fact that all
-        these properties are implied by the conjunction of just two of them:"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex
+display="inline"><![CDATA[∀M]]></fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
     </fr:p><fr:p>
-        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
-        classifier."
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
     </fr:p><fr:p>
-        "… Before that, however, we must mention that the above definition of a
-        topos, though it is appreciably simplified from that originally adopted
-        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
-        not the simplest possible; one can combine the exponentials and the
-        subobject classifier into a single structure, that of power objects."
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
-        products and all its objects are exponentiable. … Nevertheless, most of
-        the cartesian closed categories we shall meet will have equalizers as
-        well as finite products; we shall indicate this (with a note of mild
-        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
-        classifier."
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
     </fr:p><fr:p>
-        Note though, that McLarty "assumes a base category with finite limits"
-        when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="makkai-reyes-1977.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    While reading <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Okay so they only define a Grothendieck topos. We'll leave the
-        discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
+    "that's… the point"-- "realization" that first order logic is the
+    meta-language of axiomatic mathematics, just as (it seems) logical
+    frameworks are the meta-languages of (typically λ-calculus based)
+    programming languages. Well not literally, but the discrepancy is
+    non-substantial.
+</fr:p><fr:p>
+    That is, (classical) FoL is often the setting in which axiomatic
+    mathematical theories are formalized. Classic (no pun intended) examples
+    are <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{group} = \{e, ⋅, -⁻¹\}, Th(\textsf {groups}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{DLO} = \{≤\}, Th(\textsf {Dense Linear Orders}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{sets} = \{∈\}, ZFC)]]></fr:tex>.
+    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
+</fr:p><fr:p>
+    "definite property"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="awodey-2010.xml"
-type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
-        <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
-            </fr:li>
-            <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
-            </fr:li>
-            <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
-            </fr:li></fr:ul>
-        This compact definition proves to be amazingly rich in consequences:
-        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="goldblatt-2006.xml"
-type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
-        <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
-            </fr:li>
-            <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
-            </fr:li>
-            <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
-            </fr:li>
-            <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
-            </fr:li></fr:ul>
-        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
-        <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
-            </fr:li></fr:ul>
-        and dually (2) replaced by
-        <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
-            </fr:li></fr:ul>
-        The definition just given is the one originally proposed by Lawvere and
-        Tierney, in terms of which they started [elementary] topos theory in
-        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
-        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
-        a topos can be defined as a Cartesian closed category with a subobject
-        classifier. In 4.7 we shall consider a different definition, based on a
-        categorial characterisation of power sets."
+href="carette-kiselyov-shan-2007.xml"
+addr="carette-kiselyov-shan-2007"
+title="Finally Tagless, Partially Evaluated">finally tagless encodings</fr:link>
+        and <fr:link
+type="external"
+href="hoffman-1999">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
     </fr:p><fr:p>
-        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
-        complete</fr:em>, meaning every finite diagram (finite objects, finite
-        morphisms) has a limit, "constitute", meaning that "cartesian closed"
-        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        Again, quotes are air-quotes, not actual quotes.
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
-type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+type="external"
+href="sterling-2022.pdf">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
-type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
-            <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
-                </fr:li>
-                <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
-                </fr:li>
-                <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+type="external"
+href="blass-2003.pdf">blass-2003.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
+    Do you only actually need functors? Zeilberger's <fr:link
+type="external"
+href="millies-zeilberger-2015.pdf">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
+    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
+</fr:p><fr:p>
+    Talked to Max about this on 5-9.
+    They really just give to different notions of "logic over syntax"--
+    a hoare logic, and a implicational (eg FO) logic.
+    I suppose one'd want to study a bit the relationship between the two.
+    <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
-            <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
-            classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
-            any category which satisfies (1) and has all exponentials will, in
-            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
-type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
-            course of this chapter that the two sets of axioms are actually
-            equivalent.
-        </fr:p><fr:p>
-            Notice that the two natural isomorphisms (1) and (2) may be combined
-            in a single isomorphism
-            <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
-            also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
-    every definition, and give a standard one:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+href="millies-zeilberger-2015.xml"
+addr="millies-zeilberger-2015"
+title="Functors are Type Refinement Systems">paper</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
-    <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
-    is a pullback square.
-</fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
-type="local"
-addr="hejohns-000K">global
-    element</fr:link> is trivially monic. Recall that "monos are stable
-    under pullback".
-</fr:p><fr:p>
-    This is from <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
-    So comparing each definition with another, they're all about par for
-    course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Contributions</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>432</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -4076,1776 +4805,1849 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
 </fr:p><fr:p>
     Combining the two names, I think "higher-order intuitionistic topos logic"
     is at least a little more descriptive.
-    Although I like calling it "finite-type intuitionistic predicate logic" to
-    keep in touch with some older terminology you still see around the block.
-
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    There is some (a posteriori insignificant) variation in the presentation of
-    the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
-type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
-    juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
-type="local"
-addr="hejohns-000G">as done for elementary
-    topoi</fr:link>--, but this time we slightly modify the source for
-    uniformity as needed.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
-type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
-        <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
-            <fr:li>
-                a class of <fr:em>terms</fr:em> of each type, including countably many
-                variables of each type
-            </fr:li>
-            <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
-            </fr:li></fr:ul>"
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "The class of terms is freely generated from certain basic terms by
-            certain operations including the following… and is closed under certain operations.
-            <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
-                </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
-                </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
-                </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
-                </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
-        </fr:p><fr:p>
-            "One may also define a number of widely used term forming
-            operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
-                </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
-type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
-        I don't know if Johnstone prefers the term "signature" because
-        "language" is overloaded and often refers not just to the, well,
-        language, but to the larger logic sans deductive system, but that would
-        be a good reason.)
-    </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
-        of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-                (i) <fr:em>Basic types</fr:em>: Each sort is a type
-            </fr:li>
-            <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
-                type.
-            </fr:li>
-            <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
-                a type.
-            </fr:li>
-            <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
-            </fr:li>
-            <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
-            </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
-        </fr:p><fr:p>
-            "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
-        </fr:p><fr:p>
-            "From now on, all our languages will contain the product type
-            constructor; thus we have restricted all our primitive function and
-            relation symbols to be unary. (If our signature also contaisn
-            function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
-            primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
-        "The term constructor for power types in fact builds terms form
-        formulae, so over a language which contains power types we have to
-        define terms and formulae by a simultaneous recursion".
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            (We omit the bookkeeping of free variables that Johnstone inlines with
-            the following definitions.)
-            "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (iii)
-                    <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
-                        </fr:li>
-                        <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
-                        </fr:li></fr:ul></fr:li>
-                <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
-                    [This is formally different from (ii).]
-                </fr:li>
-                <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (viii)
-                    <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
-                        </fr:li>
-                        <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
-                        </fr:li></fr:ul></fr:li>
-                <fr:li>
-                    (ix) The remaining term constructor associated with list
-                    types, the <fr:em>iterator</fr:em>, is more complicated [which we
-                    omit for irrelevance].
-                </fr:li></fr:ul></fr:p><fr:p>
-            Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
-            types, and clauses (viii) and (ix) only if we have list types.
-            Note that, except for power types, the clauses come in pairs [of
-            constructors and destructors]. The reason why we have no
-            'destructor' for power types is that, just as the constructor for
-            power types builds terms out of formulae rather than terms, the
-            corresponding destructor builds formulae out of terms of a power
-            type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    Although I like calling it "finite-type intuitionistic predicate logic" to
+    keep in touch with some older terminology you still see around the block.
+
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
-            <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
-                </fr:li>
-                <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
-                </fr:li>
-                <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
-                </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>434</fr:anchor><fr:addr>hejohns-000R</fr:addr><fr:route>hejohns-000R.xml</fr:route><fr:title>BLANK</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>17</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>436</fr:anchor><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:taxon>Person</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Short About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
-        interpretation), instead of describing the logic abstractly and taking
-        the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    I am a master's student in Computer Science and Engineering at the
+    University of Michigan.
+    I'm interested in logic and related areas of computer science, particularly
+    programming languages.
+</fr:p><fr:p>
+    This is my <fr:link
+type="external"
+href="https://www.jonmsterling.com/jms-005P.xml">forest</fr:link>.
+</fr:p><fr:ul><fr:li>
+                email: <fr:embedded-tex
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble><![CDATA[]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[hejohns@umich.edu]]></fr:embedded-tex-body></fr:embedded-tex></fr:li>
+    <fr:li>
+        photo:
+        <fr:p><html:img
+xmlns:html="http://www.w3.org/1999/xhtml"
+src="hejohns.jpeg"
+width="150"
+height="150" /></fr:p></fr:li>
+    <fr:li>
+        cv: <fr:link
+type="external"
+href="cv.pdf">cv.pdf</fr:link> (TODO)
+    </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree>
+<fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#205</fr:addr><fr:route>unstable-205.xml</fr:route><fr:title>Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        My screen name is <fr:code>hejohns</fr:code>, eg <fr:link
+type="external"
+href="https://github.com/hejohns">https://github.com/hejohns</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p /></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>440</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+type="external"
+href="https://github.com/tomdjong/MGS-categorical-realizability">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "The [language] is sorted…. We inductively define terms and their sorts:
-            <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
-                </fr:li>
-                <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
-                </fr:li></fr:ul>"
-            <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
-            these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
-        </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Ignoring the natural numbers type and the idiosyncratic list type of
-        <fr:link
-href="johnstone-2002.xml"
-type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
-        elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
-        predicate logic" aka "topos logic" aka "higher-order intuitionistic
-        logic"</fr:link>, although certain inter-definable choices can be
-        made.
-    </fr:p><fr:p>
-        There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
-    </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
-<fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>442</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Some basic questions, whose "elevator pitch" answers ought to roll off
+        the tongue:
+        <fr:ul><fr:li>
+                What is a(n elementary) topos?
+            </fr:li>
+            <fr:li>
+                What is this "topos logic" that is sound in any elementary topos?
+                As in literally, what is the syntax?
+                Is it the same thing as this "higher-order intuitionistic logic"?
+                What are the inference rules/axioms?
+            </fr:li>
+            <fr:li>
+                What is this "Kripke-Joyal semantics"?
+                Is this the same thing as the so-called "forcing semantics"?
+            </fr:li>
+            <fr:li>
+                What is this "Mitchell-Bénabou language"?
+                Is it the same thing as "topos logic"?
+            </fr:li></fr:ul>
+        Following <fr:link
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+    </fr:p><fr:p>
+        Some more difficult questions we do not attempt to answer at the moment:
+        <fr:ul><fr:li>
+                What is a Grothendieck topos, and what's the difference v.
+                elementary toposes?
+            </fr:li>
+            <fr:li>
+                What is a "geometric logic" and a "geometric morphism"?
+                And what's the difference v. a "logical morphism"?
+            </fr:li>
+            <fr:li>
+                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
+                What is forcing??
+                Why does the concept of "generic" show up everywhere?
+            </fr:li>
+            <fr:li>
+                What's all this algebraic geometry stuff?
+            </fr:li>
+            <fr:li>
+                And everything else…
+            </fr:li></fr:ul></fr:p><fr:p>
+        Although as you might expect, we quickly derail from "elevator pitch"
+        answers, to "quick, write that down before I lose motivation and never
+        complete the sentence".
+        A truly FAQ-style answer compendium will have to come later-- after I've
+        actually learned the basics.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
-    </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+type="local"
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
+type="local"
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Notice that <fr:link
-href="sub-0006.xml"
-type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
-        Thus <fr:link
-href="hejohns-0006.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
-href="hejohns-0006.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
-        immediately natural question, but can be easily strengthened to
-        the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Mentioned in passing as <fr:link
-href="street-2012.xml"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
+        classifier and a natural numbers object."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
-        </fr:li>
-        <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
-    This is conventionally left implicit, but we remark to clarify what we mean
-    by the family of natural isomorphisms being natural.
-</fr:p><fr:p>
-    NOTE: I couldn't get よ to render in the latex (diagrams).
-    I'm not sure if it's a lualatex/dvi or font issue.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    We take inspiration from <fr:link
-href="hejohns-0006.xml"
+href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
-    That is, we have two goals:
-    <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
-        </fr:li>
-        <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
-        </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
-    component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
-</fr:p><fr:ul><fr:li>
-        (∃)
-        <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
-        </fr:p>
-        <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
-        </fr:p>
-        <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
-            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
-            difficult without the clarifying perspective of the Yoneda
-            lemma (no, seriously, although I'm sure one could do it).
-        </fr:p>
-        <fr:p>
-            It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
-            composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
-            Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
-            The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
-        </fr:p>
-        <fr:p>
-            Notice that the <fr:link
-href="subb-000D.xml"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
-        <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
-        </fr:p>
-        <fr:ul><fr:li><fr:p>
-                    Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
-                    determines (ie freely extends to) a natural
-                    transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
-            <fr:li><fr:p>
-                    Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
-                    embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
-        <fr:p>
-            With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
-            square</fr:link>.
-            <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
-            by the Yoneda Lemma.
-            (<fr:link
-href="subbb-000D.xml"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
-            circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
-            more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
-            composition, we have enough to tools consider
-            functoriality.
-        </fr:p>
-        <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
-            <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
-        </fr:p>
-        <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            it is indeed the case.
-        </fr:p></fr:li>
-    <fr:li>
-        (!)
-        <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
-    So this result is really a corollary of the Yoneda lemma.
-    <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+type="local"
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    However, unlike <fr:link
-href="hejohns-0006.xml"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
-    so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course, if we ignore natural numbers objects for now.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Many authors object to the name "higher-order intuitionistic logic".
+</fr:p><fr:p><fr:link
+type="local"
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+    intuitionistic' makes little sense, since it is not clear that
+    constructions like the power set are intuitionistically acceptable. What is
+    meant by the phrase is a type theory built on a formally intuitionistic
+    underlying logic." (p. 2).
+</fr:p><fr:p>
+    And <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+    axioms are used to define the higher order internal language. The logic is
+    called 'topos logic' here, although 'intuitionistic logic' is often
+    referred to in the literature. I believe that 'intuitionism' is usually,
+    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
+    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
+    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
+    (1979). But topos logic coincides with no intuitionistic logic studied
+    before toposes, and this is due to real philosophical differences (see
+    McLarty 1990a)." (p. vii).
+</fr:p><fr:p>
+    Of course this discrepancy is what one should come to expect, since the
+    very notion of a formalized logic of intuitionistic reasoning is at odds
+    with Brouwer's notion of intuitionism, so when one speaks of
+    "intuitionistic logic" it's usually in the purely formal sense.
+</fr:p><fr:p>
+    Combining the two names, I think "higher-order intuitionistic topos logic"
+    is at least a little more descriptive.
+    Although I like calling it "finite-type intuitionistic predicate logic" to
+    keep in touch with some older terminology you still see around the block.
+
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
+    
+    <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    There is some (a posteriori insignificant) variation in the presentation of
+    the syntax, thus following through with some input from <fr:link
 type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
-    </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
+    juxtapose multiple definitions-- <fr:link
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
+    topoi</fr:link>--, but this time we slightly modify the source for
+    uniformity as needed.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
+        <fr:ul><fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
+            <fr:li>
+                a class of <fr:em>terms</fr:em> of each type, including countably many
+                variables of each type
+            </fr:li>
+            <fr:li>
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
+            </fr:li></fr:ul>"
+    </fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The class of types is closed under the inductive clauses:
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The class of terms is freely generated from certain basic terms by
+            certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
+                </fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
-type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
+        </fr:p><fr:p>
+            "One may also define a number of widely used term forming
+            operations: …
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (We write "language" whenever Johnstone writes "signature", and use the
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
+        I don't know if Johnstone prefers the term "signature" because
+        "language" is overloaded and often refers not just to the, well,
+        language, but to the larger logic sans deductive system, but that would
+        be a good reason.)
+    </fr:p><fr:p>
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        of <fr:em>type constructors</fr:em> chosen from the following list, which
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
+    </fr:p><fr:tree
+toc="true"
 numbered="true"
-show-heading="false"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
-    course text(s). I haven't used it much for reading either, owing to the
-    large amount of new notation one has to refresh every time you open the
-    book, but it's a solid reference book.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
-    title, and Makkai completeness.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    "Bibliographical Note: This Dover edition, first published in 2006, is a
-    slightly corrected unabridged republication of the revised (second) edition
-    of the work originally published in 1984 by Elsevier Science Publishers
-    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
-    "Studies in Logic and the Foundations of Mathematics." A new Preface and
-    updated Bibliography have been specially prepared for this reprint."
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="mclarty-1990a.pdf"
-type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+                (i) <fr:em>Basic types</fr:em>: Each sort is a type
+            </fr:li>
+            <fr:li>
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
+                type.
+            </fr:li>
+            <fr:li>
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
+                a type.
+            </fr:li>
+            <fr:li>
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
+            </fr:li>
+            <fr:li>
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
+            </fr:li></fr:ul><fr:p>
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
+        </fr:p><fr:p>
+            "If our language has the type constructors (ii) and (iv), we shall
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
+        </fr:p><fr:p>
+            "From now on, all our languages will contain the product type
+            constructor; thus we have restricted all our primitive function and
+            relation symbols to be unary. (If our signature also contaisn
+            function types, then we can further restrict our function symbols
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
+            primitive relation symbols by constants as well)"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+        "The term constructor for power types in fact builds terms form
+        formulae, so over a language which contains power types we have to
+        define terms and formulae by a simultaneous recursion".
+    </fr:p><fr:tree
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            (We omit the bookkeeping of free variables that Johnstone inlines with
+            the following definitions.)
+            "<fr:ul><fr:li>
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (iii)
+                    <fr:ul><fr:li>
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
+                        </fr:li>
+                        <fr:li>
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
+                        </fr:li></fr:ul></fr:li>
+                <fr:li>
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
+                    [This is formally different from (ii).]
+                </fr:li>
+                <fr:li>
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (viii)
+                    <fr:ul><fr:li>
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
+                        </fr:li>
+                        <fr:li>
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
+                        </fr:li></fr:ul></fr:li>
+                <fr:li>
+                    (ix) The remaining term constructor associated with list
+                    types, the <fr:em>iterator</fr:em>, is more complicated [which we
+                    omit for irrelevance].
+                </fr:li></fr:ul></fr:p><fr:p>
+            Of course, clauses (v) and (vi) of the definition above apply only
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
+            types, and clauses (viii) and (ix) only if we have list types.
+            Note that, except for power types, the clauses come in pairs [of
+            constructors and destructors]. The reason why we have no
+            'destructor' for power types is that, just as the constructor for
+            power types builds terms out of formulae rather than terms, the
+            corresponding destructor builds formulae out of terms of a power
+            type-- it thus appears as clause (iii) of the next definition."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The unity of opposites in the title is essentially that between logic
-        and geometry, and there are compelling reasons for maintaining that
-        geometry is the leading aspect. At the same lime, in the present joint
-        work with Myles Tierney there are important influences in the other
-        direction: a Grothendieck "topology" appears most naturally as a
-        modal operator, of the nature "it is locally the case that", the
-        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
-        apply to families of geometrical objects rather than to propositional
-        functions, and an important technique is to lift constructions first
-        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
-        topos. We first sum up the principal contradictions of the
-        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
-        adjoint functors, significantly generalizing the theory to free it of
-        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
-        enabling one to claim that in a sense logic is a special case of
-        geometry). The method thus developing is then applied to intrinsically
-        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
-        the independence of the continuum hypothesis free of any use of
-        transfinite induction. The second application of the method outlined
-        here is an intrinsic geometric construction of the Chevalley-Hakim
-        global spectrum of a ringed topos free of any choice of a "site of
-        definition".
-    </fr:p><fr:p>
-        When the main contradictions of a thing have been found,
-        the scientific procedure is to summarize them in slogans which one then
-        constantly uses as an ideological weapon for the further development
-        and transformation of the thing. Doing this for "set theory" requires
-        taking account of the experience that the main pairs of opposing
-        tendencies in mathematics take the form of adjoint functors, and frees
-        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
-        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
-        metaphysical "construction". Further, experience with sheaves,
-        permutation representations, algebraic spaces, etc., shows that a "set
-        theory" for geometry should apply not only to abstract sets divorced
-        from time, space, ring of definition, etc., but also to more general
-        sets which do in fact develop along such parameters. For such sets,
-        usually logic is "intuitionistic" (in its formal properties)
-        usually the axiom of choice is false, and usually a set is not
-        determined by its points defined over <fr:tex>1</fr:tex> only.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    I have not read this, and have no idea what "contradictions" he's referring to.
-    <fr:link
-href="lawvere-1970.pdf"
-type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            (We again omit the bookkeeping of free variables.)
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
+            <fr:ul><fr:li>
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
+                </fr:li>
+                <fr:li>
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
+                </fr:li>
+                <fr:li>
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
+                </fr:li></fr:ul></fr:p><fr:p>
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
+        interpretation), instead of describing the logic abstractly and taking
+        the internal language to be a diagram (in Tarskian-model theory speak)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+    </fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The view that toposes originated as generalized set theory is a figment of
-        set theoretically educated common sense. This false history obstructs
-        understanding of category theory and especially of categorical foundations
-        for mathematics. Problems in geometry, topology, and related algebra led to
-        categories and toposes. Elementary toposes arose when Lawvere's interest in
-        the foundations of physics and Tierney's in the foundations of topology led
-        both to study Grothendieck's foundations for algebraic geometry. I end with
-        remarks on a categorical view of the history of set theory, including a
-        false history plausible from that point of view that would make it helpful
-        to introduce toposes as a generalization from set theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="mclarty-1990b.pdf"
-type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I have not read this. This might be a good source, aside from <fr:link
-href="ronnie-chen.xml"
-type="local"
-addr="ronnie-chen">Ronnie Chen</fr:link>,
-    where the viewpoint of "generalized algebraic semantics" is developed.
-</fr:p><fr:p><fr:link
-href="marquis-reyes-2012.pdf"
-type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            The types have the structure of the objects of an elementary topos,
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The [language] is sorted…. We inductively define terms and their sorts:
+            <fr:ul><fr:li>
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
+                </fr:li>
+                <fr:li>
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
+                </fr:li></fr:ul>"
+            <fr:ul><fr:li>
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
+            these terms in any elementary topos, developed in 13.3 and 13.4],
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
+        </fr:p><fr:p>
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Ignoring the natural numbers type and the idiosyncratic list type of
+        <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+        elementary topops for reference) of <fr:link
+type="local"
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
+        predicate logic" aka "topos logic" aka "higher-order intuitionistic
+        logic"</fr:link>, although certain inter-definable choices can be
+        made.
+    </fr:p><fr:p>
+        There's no clear notion of "taking everything", but a reasonable
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
+    </fr:p><fr:p>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
+<fr:tree
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    One of the early NBG papers. Maybe "the" NBG paper.
-    My notes on this are currently in <fr:link
-href="quine-1960.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Basic information</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    I am a master's student in Computer Science and Engineering at the
+    University of Michigan.
+    I'm interested in logic and related areas of computer science, particularly
+    programming languages.
 </fr:p><fr:p>
-    English translation in <fr:link
-href="van-heijenoort-1967.xml"
-type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+    This is my <fr:link
+type="external"
+href="https://www.jonmsterling.com/jms-005P.xml">forest</fr:link>.
+</fr:p><fr:ul><fr:li>
+                email: <fr:embedded-tex
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble><![CDATA[]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[hejohns@umich.edu]]></fr:embedded-tex-body></fr:embedded-tex></fr:li>
+    <fr:li>
+        photo:
+        <fr:p><html:img
+xmlns:html="http://www.w3.org/1999/xhtml"
+src="hejohns.jpeg"
+width="150"
+height="150" /></fr:p></fr:li>
+    <fr:li>
+        cv: <fr:link
+type="external"
+href="cv.pdf">cv.pdf</fr:link> (TODO)
+    </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
-    this list. Whatever else you might think about Quine, he was a terrific
-    explainer, and this is the best introduction to the subject I know."
-</fr:p><fr:p>
-    (Yes, that 50 should be a 60(b).)
-    Here is the followup I wrote:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
-    </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:addr>hejohns-000N</fr:addr><fr:route>hejohns-000N.xml</fr:route><fr:title>L (Language)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    I vaguely recall reading that the tradition of isolating the non-logical
+    symbols (constants, functions, relations) of a first order logic for a
+    language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, and referring to suitable models as
+    <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>-structures, was popularized by Carnap, and maybe in
+    particular by <fr:em>The Logical Syntax of Language</fr:em>.
+</fr:p><fr:p>
+    I am unable to track down a source for this now. I'll have to check <fr:em>The
+    Syntax</fr:em> sometime. I can say at least <fr:em>Meaning and Necessity</fr:em> does
+    <fr:em>not</fr:em> make use of this terminology in the same way.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>448</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+type="local"
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We have built the first family of tagless interpretations for a
-        higher-order typed object language in a typed metalanguage (Haskell or
-        ML) that require no dependent types, generalized algebraic data types,
-        or postprocessing to eliminate tags. The statically type-preserving
-        interpretations include an evaluator, a compiler (or staged evaluator),
-        a partial evaluator, and call-by-name and call-by-value CPS
-        transformers.
-    </fr:p><fr:p>
-        Our main idea is to encode HOAS using cogen functions rather than data
-        constructors. In other words, we represent object terms not in an
-        initial algebra but using the coalgebraic structure of the λ-calculus.
-        Our representation also simulates inductive maps from types to types,
-        which are required for typed partial evaluation and CPS
-        transformations.
-    </fr:p><fr:p>
-        Our encoding of an object term abstracts over the various ways to
-        interpret it, yet statically assures that the interpreters never get
-        stuck.  To achieve self-interpretation and show Jones-optimality, we
-        relate this exemplar of higher-rank and higher-kind polymorphism to
-        plugging a term into a context of let-polymorphic bindings.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
+        classifier and a natural numbers object."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the APLAS version (conference paper). There is a roughly twice
-        as long journal version.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
     </fr:p><fr:p>
-        I didn't study a lot of the probably more interesting aspects of this
-        work.
-        I just wanted a first/resonant-understanding of "what does <fr:em>finally
-        tagless</fr:em> mean?".
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
     </fr:p><fr:p>
-        It seems the very basic idea is about the general problem of DSLs.
-        The very short way to say it, is that we like to construct languages as
-        object languages of nice meta languages.
-        For example, a nice meta languages gives us nice primitives to reuse as
-        the denotational semantics of the object language.
-        A classic example is to use the <fr:code>Int</fr:code> type in the meta language as
-        the semantics of the DSL/embedded/object language's numeric type.
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        Most people's knee jerk reaction is to encode the object language as an
-        ADT. This is fine and well, and can be called <fr:em>initial</fr:em> in the sense
-        that it is the free syntax generated by the encoded rules/initial
-        algebra/inductive dataype.
-        But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
+href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
     </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
-        sympathy to this line of thought, I’m skeptical of the unspoken assumption
-        that there’s an underlying concept of a ‘foundation’ up for analysis, that
-        this analysis would properly guide our assessment of the various
-        candidates. In contrast, it seems to me that the considerations the
-        combatants offer against opponents and for their preferred candidates, as
-        well as the roles each candidate actually or potentially succeeds in
-        playing, reveal quite a number of different jobs that mathematicians want
-        done. What matters is these jobs we want our theories to do and how well
-        they do them. Whether any or all of them, jobs or theories, deserves to be
-        called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    A nice article that can be briefly summarized as
-    <fr:ul><fr:li>
-            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
-            system, whatever that means
-        </fr:li>
-        <fr:li>
-            Classical set theory is perfectly fine as a foundational system in
-            the traditionally (ie ZFC) conceived sense
-        </fr:li>
-        <fr:li>
-            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
-            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
-            bleeding edge").
-        </fr:li></fr:ul></fr:p><fr:p>
-    The contrast between the minimalistic ZFC and potentially enormous "user
-    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
-    natural deduction. Indeed, it's like how Ronnie calls set theory "the
-    machine/assembly language of math" (kind of paraphrase), and Max calls SK
-    combinators "the machine/assembly language of λ-calculus".
-</fr:p><fr:p>
-    Natural deduction is certainly more complicated than SK/Hilbert proof
-    systems, but has nice usability properties that are worth study and
-    implementation.
-    The general mathematical trend seems to be that of starting with a spartan
-    formal system, then gradually making it more user friendly to accomodate
-    every increasingly complicated mathematics, where the minimalistic system
-    starts to leave one wanting.
-</fr:p><fr:p>
-    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
-    redundant to mention other proof assistants eg Cubical Agda.
-</fr:p><fr:p>
-    I've used quotes here as air quotes, not actual quotes of Maddy.
-    <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
 type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+type="local"
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
-    </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
-        <fr:ul><fr:li>
-                completness theorems,
-            </fr:li>
-            <fr:li>
-                the solvability of higher type functional equations, and
-            </fr:li>
-            <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course, if we ignore natural numbers objects for now.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>450</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="local"
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+    (Recall that categories can be formalized as a two or dependently-sorted
+    first order theory.)
+</fr:p><fr:p>
+    This is corroborated by <fr:link
+type="local"
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+    that followed, Lawvere tried to extend his analysis and sketched a
+    categorical version of first-order theories under the name of elementary
+    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
+    introduced the notion of an elementary topos, making an explicit connection
+    with higher-order logic and type theories. Both Lawvere and Tierney were
+    aiming at an elementary, that is first-order, axiomatic presentation of
+    what are now called Grothendieck toposes, a special type of categories
+    introduced by Alexandre Grothendieck in the context of algebraic geometry
+    and sheaf theory. Soon after, connections with intuitionistic analysis,
+    recursive functions, completeness theorems for various logical systems,
+    differential geometry, constructive mathematics were made." (p. 2).
+</fr:p><fr:p>
+    If <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+    it's curious to note that the word "elementary" never appears.
+    It must've been coined sometime later, but I don't know the details.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>452</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Fibered Right Adjoints are automatically fibered?
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -5861,18 +6663,20 @@ root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:taxon>Abstract</fr:ta
         present a few applications. As a larger case study, we revisit
         Reynolds’ paper on “The Meaning of Types” (2000), showing how the
         paper’s main results may be reconstructed along these lines.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex
+display="inline"><![CDATA[∀M]]></fr:tex> statements.
         Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
         functor into) the syntax, typically trivially constructed as a pullback
         (it's hard in general to concoct models with structure preserving maps
@@ -5924,1388 +6728,2353 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-z
     </fr:p><fr:p>
         Maybe I'm finally close to being able to understand what a Kripke
         logical relation is.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    The first point:
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>455</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    While reading <fr:link
+type="local"
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
+    "that's… the point"-- "realization" that first order logic is the
+    meta-language of axiomatic mathematics, just as (it seems) logical
+    frameworks are the meta-languages of (typically λ-calculus based)
+    programming languages. Well not literally, but the discrepancy is
+    non-substantial.
+</fr:p><fr:p>
+    That is, (classical) FoL is often the setting in which axiomatic
+    mathematical theories are formalized. Classic (no pun intended) examples
+    are <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{group} = \{e, ⋅, -⁻¹\}, Th(\textsf {groups}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{DLO} = \{≤\}, Th(\textsf {Dense Linear Orders}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{sets} = \{∈\}, ZFC)]]></fr:tex>.
+    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
+</fr:p><fr:p>
+    "definite property"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
+    Do you only actually need functors? Zeilberger's <fr:link
+type="external"
+href="millies-zeilberger-2015.pdf">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
+    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
+</fr:p><fr:p>
+    Talked to Max about this on 5-9.
+    They really just give to different notions of "logic over syntax"--
+    a hoare logic, and a implicational (eg FO) logic.
+    I suppose one'd want to study a bit the relationship between the two.
+    <fr:link
+type="local"
+href="millies-zeilberger-2015.xml"
+addr="millies-zeilberger-2015"
+title="Functors are Type Refinement Systems">paper</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
+    this list. Whatever else you might think about Quine, he was a terrific
+    explainer, and this is the best introduction to the subject I know."
+</fr:p><fr:p>
+    (Yes, that 50 should be a 60(b).)
+    Here is the followup I wrote:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        During my Topology lectures at Macquarie University in the 1970s, the
-        students and I realized, in proving that a function f between posets was
-        order preserving when there was a function u in the reverse direction such
-        that f (x) ≤ a if and only if x ≤ u(a), did not require u to be order
-        preserving. I realized then that knowing functors in the two directions
-        only on objects and the usual hom adjointness isomorphism implied the
-        effect of the functors on homs was uniquely determined. Writing this down
-        properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
+    </fr:p><fr:p>
+        I was surprised by the mention of von Neumann 1925 in this snippet:
+    </fr:p><fr:p>
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
+    </fr:p><fr:p>
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
+    </fr:p><fr:p>
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+type="external"
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem">class existence theorem</fr:link>.
+    </fr:p><fr:p>
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
+    </fr:p><fr:p>
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:ul><fr:li><fr:link
+type="external"
+href="quine-1960.pdf">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
+type="local"
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>460</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+type="external"
+href="zeilberger">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>462</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
-<fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        TODO: not a good selection
-    </fr:p><fr:ul><fr:li><fr:link
-href="https://github.com/eecs490/gradescope-utils"
-type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
-        <fr:li><fr:link
-href="https://github.com/hejohns/.rc/tree/master/home/bin"
-type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        <fr:li>
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://mplse-reading-group.github.io/"
-type="external">https://mplse-reading-group.github.io/</fr:link></fr:li>
-        <fr:li><fr:link
-href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
-type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
-href="../posts.html"
-type="external">Jekyll Blog</fr:link></fr:mainmatter></fr:tree>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="true"
-show-heading="false"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>443</fr:anchor></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html"
-type="external">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
-href="stack-overflow/every-topos-is-a-regular-category.html"
-type="external">stack-overflow/every-topos-is-a-regular-category.html</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
-    title, and Makkai completeness.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The unity of opposites in the title is essentially that between logic
-        and geometry, and there are compelling reasons for maintaining that
-        geometry is the leading aspect. At the same lime, in the present joint
-        work with Myles Tierney there are important influences in the other
-        direction: a Grothendieck "topology" appears most naturally as a
-        modal operator, of the nature "it is locally the case that", the
-        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
-        apply to families of geometrical objects rather than to propositional
-        functions, and an important technique is to lift constructions first
-        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
-        topos. We first sum up the principal contradictions of the
-        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
-        adjoint functors, significantly generalizing the theory to free it of
-        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
-        enabling one to claim that in a sense logic is a special case of
-        geometry). The method thus developing is then applied to intrinsically
-        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
-        the independence of the continuum hypothesis free of any use of
-        transfinite induction. The second application of the method outlined
-        here is an intrinsic geometric construction of the Chevalley-Hakim
-        global spectrum of a ringed topos free of any choice of a "site of
-        definition".
-    </fr:p><fr:p>
-        When the main contradictions of a thing have been found,
-        the scientific procedure is to summarize them in slogans which one then
-        constantly uses as an ideological weapon for the further development
-        and transformation of the thing. Doing this for "set theory" requires
-        taking account of the experience that the main pairs of opposing
-        tendencies in mathematics take the form of adjoint functors, and frees
-        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
-        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
-        metaphysical "construction". Further, experience with sheaves,
-        permutation representations, algebraic spaces, etc., shows that a "set
-        theory" for geometry should apply not only to abstract sets divorced
-        from time, space, ring of definition, etc., but also to more general
-        sets which do in fact develop along such parameters. For such sets,
-        usually logic is "intuitionistic" (in its formal properties)
-        usually the axiom of choice is false, and usually a set is not
-        determined by its points defined over <fr:tex>1</fr:tex> only.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    I have not read this, and have no idea what "contradictions" he's referring to.
-    <fr:link
-href="lawvere-1970.pdf"
-type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I have not read this. This might be a good source, aside from <fr:link
-href="ronnie-chen.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="ronnie-chen">Ronnie Chen</fr:link>,
-    where the viewpoint of "generalized algebraic semantics" is developed.
-</fr:p><fr:p><fr:link
-href="marquis-reyes-2012.pdf"
-type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="subb-000D.xml"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Some basic questions, whose "elevator pitch" answers ought to roll off
-        the tongue:
-        <fr:ul><fr:li>
-                What is a(n elementary) topos?
-            </fr:li>
-            <fr:li>
-                What is this "topos logic" that is sound in any elementary topos?
-                As in literally, what is the syntax?
-                Is it the same thing as this "higher-order intuitionistic logic"?
-                What are the inference rules/axioms?
-            </fr:li>
-            <fr:li>
-                What is this "Kripke-Joyal semantics"?
-                Is this the same thing as the so-called "forcing semantics"?
-            </fr:li>
-            <fr:li>
-                What is this "Mitchell-Bénabou language"?
-                Is it the same thing as "topos logic"?
-            </fr:li></fr:ul>
-        Following <fr:link
-href="mclarty-1992.xml"
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
-    </fr:p><fr:p>
-        Some more difficult questions we do not attempt to answer at the moment:
-        <fr:ul><fr:li>
-                What is a Grothendieck topos, and what's the difference v.
-                elementary toposes?
-            </fr:li>
-            <fr:li>
-                What is a "geometric logic" and a "geometric morphism"?
-                And what's the difference v. a "logical morphism"?
-            </fr:li>
-            <fr:li>
-                In what sense is "Kripke-Joyal semantics" a "forcing semantics"?
-                What is forcing??
-                Why does the concept of "generic" show up everywhere?
-            </fr:li>
-            <fr:li>
-                What's all this algebraic geometry stuff?
-            </fr:li>
-            <fr:li>
-                And everything else…
-            </fr:li></fr:ul></fr:p><fr:p>
-        Although as you might expect, we quickly derail from "elevator pitch"
-        answers, to "quick, write that down before I lose motivation and never
-        complete the sentence".
-        A truly FAQ-style answer compendium will have to come later-- after I've
-        actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="subbb-000D.xml"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="blass-1988.xml"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    However, unlike <fr:link
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
-    (Recall that categories can be formalized as a two or dependently-sorted
-    first order theory.)
-</fr:p><fr:p>
-    This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>464</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
-    that followed, Lawvere tried to extend his analysis and sketched a
-    categorical version of first-order theories under the name of elementary
-    theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
-    introduced the notion of an elementary topos, making an explicit connection
-    with higher-order logic and type theories. Both Lawvere and Tierney were
-    aiming at an elementary, that is first-order, axiomatic presentation of
-    what are now called Grothendieck toposes, a special type of categories
-    introduced by Alexandre Grothendieck in the context of algebraic geometry
-    and sheaf theory. Soon after, connections with intuitionistic analysis,
-    recursive functions, completeness theorems for various logical systems,
-    differential geometry, constructive mathematics were made." (p. 2).
-</fr:p><fr:p>
-    If <fr:link
-href="lawvere-1970.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
-    it's curious to note that the word "elementary" never appears.
-    It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
+        <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
-    Of course, these are always the kind of descriptions that only make sense
-    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
-    We remark that <fr:link
-href="mclarty-1990b.xml"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
-    perspective.
-</fr:p><fr:p>
-    Instead, I've gathered every book I could conveniently get my hands on just
-    now, and we'll see how each defines an elementary topos.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
-        classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "We are now ready to present the definition of an (elementary) topos,
-        which is the central notion of this book: it is a category having all
-        the good 'set-like' properties discussed in the previous chapter,
-        except (possibly) for Booleanness. It is a remarkable fact that all
-        these properties are implied by the conjunction of just two of them:"
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
+type="local"
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
     </fr:p><fr:p>
-        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
-        classifier."
+        This proof is rather low tech and is lifted straight from <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
     </fr:p><fr:p>
-        "… Before that, however, we must mention that the above definition of a
-        topos, though it is appreciably simplified from that originally adopted
-        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
-        not the simplest possible; one can combine the exponentials and the
-        subobject classifier into a single structure, that of power objects."
+        The forward directions are trivial.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
+        Then <fr:link
+type="local"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+type="local"
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
-        products and all its objects are exponentiable. … Nevertheless, most of
-        the cartesian closed categories we shall meet will have equalizers as
-        well as finite products; we shall indicate this (with a note of mild
-        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+type="local"
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+type="local"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
+                    <fr:embedded-tex
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
-        classifier."
-    </fr:p><fr:p>
-        Note though, that McLarty "assumes a base category with finite limits"
-        when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="subb-0006.xml"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>466</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="makkai-reyes-1977.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Okay so they only define a Grothendieck topos. We'll leave the
-        discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="awodey-2010.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
-        <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
-            </fr:li>
-            <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
-            </fr:li>
-            <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
-            </fr:li></fr:ul>
-        This compact definition proves to be amazingly rich in consequences:
-        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="goldblatt-2006.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
-        <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
-            </fr:li>
-            <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
-            </fr:li>
-            <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
-            </fr:li>
-            <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
-            </fr:li></fr:ul>
-        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
-        <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
-            </fr:li></fr:ul>
-        and dually (2) replaced by
-        <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
-            </fr:li></fr:ul>
-        The definition just given is the one originally proposed by Lawvere and
-        Tierney, in terms of which they started [elementary] topos theory in
-        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
-        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
-        a topos can be defined as a Cartesian closed category with a subobject
-        classifier. In 4.7 we shall consider a different definition, based on a
-        categorial characterisation of power sets."
-    </fr:p><fr:p>
-        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
-        complete</fr:em>, meaning every finite diagram (finite objects, finite
-        morphisms) has a limit, "constitute", meaning that "cartesian closed"
-        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
-        <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
+        Thus <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>468</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="subb-000D.xml"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
-            <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
-                </fr:li>
-                <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
-                </fr:li>
-                <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+type="local"
+href="subbb-000D.xml"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
-href="maclane-moerdijk-1994.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
-            classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
-            any category which satisfies (1) and has all exponentials will, in
-            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
-type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
-            course of this chapter that the two sets of axioms are actually
-            equivalent.
-        </fr:p><fr:p>
-            Notice that the two natural isomorphisms (1) and (2) may be combined
-            in a single isomorphism
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
-            also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
-    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
-    every definition, and give a standard one:
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
-    <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
-    is a pullback square.
-</fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-000K">global
-    element</fr:link> is trivially monic. Recall that "monos are stable
-    under pullback".
-</fr:p><fr:p>
-    This is from <fr:link
-href="mclarty-1992.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
-    So comparing each definition with another, they're all about par for
-    course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Many authors object to the name "higher-order intuitionistic logic".
-</fr:p><fr:p><fr:link
-href="blass-1988.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
-    intuitionistic' makes little sense, since it is not clear that
-    constructions like the power set are intuitionistically acceptable. What is
-    meant by the phrase is a type theory built on a formally intuitionistic
-    underlying logic." (p. 2).
-</fr:p><fr:p>
-    And <fr:link
-href="mclarty-1992.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
-    axioms are used to define the higher order internal language. The logic is
-    called 'topos logic' here, although 'intuitionistic logic' is often
-    referred to in the literature. I believe that 'intuitionism' is usually,
-    and rightly, taken to mean Brouwer's epistemology of mathematics, which is
-    unrelated to the origin or content of topos theory. Topos logic strikingly resembles formal intuitionistic logic and the two have interacted, as in
-    van der Hoeven and Moerdijk (1984) or many papers in Fourman <fr:em>et al.</fr:em>
-    (1979). But topos logic coincides with no intuitionistic logic studied
-    before toposes, and this is due to real philosophical differences (see
-    McLarty 1990a)." (p. vii).
-</fr:p><fr:p>
-    Of course this discrepancy is what one should come to expect, since the
-    very notion of a formalized logic of intuitionistic reasoning is at odds
-    with Brouwer's notion of intuitionism, so when one speaks of
-    "intuitionistic logic" it's usually in the purely formal sense.
-</fr:p><fr:p>
-    Combining the two names, I think "higher-order intuitionistic topos logic"
-    is at least a little more descriptive.
-    Although I like calling it "finite-type intuitionistic predicate logic" to
-    keep in touch with some older terminology you still see around the block.
-
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
+    </fr:p><fr:p>
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    There is some (a posteriori insignificant) variation in the presentation of
-    the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
-type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
-    juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-000G">as done for elementary
-    topoi</fr:link>--, but this time we slightly modify the source for
-    uniformity as needed.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
-        <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
-            <fr:li>
-                a class of <fr:em>terms</fr:em> of each type, including countably many
-                variables of each type
-            </fr:li>
-            <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
-            </fr:li></fr:ul>"
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Notice that <fr:link
+type="local"
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
+        Thus <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
+        immediately natural question, but can be easily strengthened to
+        the form of this corollary.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Mentioned in passing as <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "The class of terms is freely generated from certain basic terms by
-            certain operations including the following… and is closed under certain operations.
-            <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
-                </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
-                </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
-                </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
-                </fr:li>
-                <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
-                </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
+        </fr:li>
+        <fr:li>
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
+    This is conventionally left implicit, but we remark to clarify what we mean
+    by the family of natural isomorphisms being natural.
+</fr:p><fr:p>
+    NOTE: I couldn't get よ to render in the latex (diagrams).
+    I'm not sure if it's a lualatex/dvi or font issue.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
-        </fr:p><fr:p>
-            "One may also define a number of widely used term forming
-            operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
-                </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
-        I don't know if Johnstone prefers the term "signature" because
-        "language" is overloaded and often refers not just to the, well,
-        language, but to the larger logic sans deductive system, but that would
-        be a good reason.)
-    </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
-        of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-                (i) <fr:em>Basic types</fr:em>: Each sort is a type
-            </fr:li>
-            <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
-                type.
-            </fr:li>
-            <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
-                a type.
-            </fr:li>
-            <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
-            </fr:li>
-            <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
-            </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
-        </fr:p><fr:p>
-            "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
-        </fr:p><fr:p>
-            "From now on, all our languages will contain the product type
-            constructor; thus we have restricted all our primitive function and
-            relation symbols to be unary. (If our signature also contaisn
-            function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
-            primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
-        "The term constructor for power types in fact builds terms form
-        formulae, so over a language which contains power types we have to
-        define terms and formulae by a simultaneous recursion".
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            (We omit the bookkeeping of free variables that Johnstone inlines with
-            the following definitions.)
-            "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (iii)
-                    <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
-                        </fr:li>
-                        <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
-                        </fr:li></fr:ul></fr:li>
-                <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
-                    [This is formally different from (ii).]
-                </fr:li>
-                <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (viii)
-                    <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
-                        </fr:li>
-                        <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
-                        </fr:li></fr:ul></fr:li>
-                <fr:li>
-                    (ix) The remaining term constructor associated with list
-                    types, the <fr:em>iterator</fr:em>, is more complicated [which we
-                    omit for irrelevance].
-                </fr:li></fr:ul></fr:p><fr:p>
-            Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
-            types, and clauses (viii) and (ix) only if we have list types.
-            Note that, except for power types, the clauses come in pairs [of
-            constructors and destructors]. The reason why we have no
-            'destructor' for power types is that, just as the constructor for
-            power types builds terms out of formulae rather than terms, the
-            corresponding destructor builds formulae out of terms of a power
-            type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    We take inspiration from <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
+    That is, we have two goals:
+    <fr:ul><fr:li>
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+        </fr:li>
+        <fr:li>
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
+        </fr:li></fr:ul>
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
+    component of the unit of the (a posteriori) adjunction
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
+</fr:p><fr:ul><fr:li>
+        (∃)
+        <fr:p>
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
+        </fr:p>
+        <fr:p>
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
-            <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
-                </fr:li>
-                <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
-                </fr:li>
-                <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
-                </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
+        </fr:p>
+        <fr:p>
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
+            indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
+            difficult without the clarifying perspective of the Yoneda
+            lemma (no, seriously, although I'm sure one could do it).
+        </fr:p>
+        <fr:p>
+            It's instructive to try directly, if one hasn't.
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
+            composition of morphisms in any diagram-- how do you reason
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
+            Instead, we want to invoke a representation theorem and work
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
+            The solution is to follow the Yoneda principle and
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
+        </fr:p>
+        <fr:p>
+            Notice that the <fr:link
+type="local"
+href="subb-000D.xml"
+addr="subb-000D">above square</fr:link> is
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
-        interpretation), instead of describing the logic abstractly and taking
-        the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
+        <fr:p>
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
+        </fr:p>
+        <fr:ul><fr:li><fr:p>
+                    Then the Yoneda lemma tells us that by specifying an
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
+                    determines (ie freely extends to) a natural
+                    transformation <fr:tex
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
+            <fr:li><fr:p>
+                    Equivalently, the Yoneda principle tells us that an
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
+                    embeds as a natural transformation <fr:tex
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
+        <fr:p>
+            With this in mind, we notice that by definition,
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
+type="local"
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
+            square</fr:link>.
+            <fr:embedded-tex
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
+            by the Yoneda Lemma.
+            (<fr:link
+type="local"
+href="subbb-000D.xml"
+addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
+            circumlocutious and you might as well start by saying such a
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
+            more obvious starting point without presupposing it exists.)
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
+            composition, we have enough to tools consider
+            functoriality.
+        </fr:p>
+        <fr:p>
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            <fr:embedded-tex
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            A diagram chase verifies that the diagram indeed commutes, and
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
+        </fr:p>
+        <fr:p>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
+            <fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            it is indeed the case.
+        </fr:p></fr:li>
+    <fr:li>
+        (!)
+        <fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
+    So this result is really a corollary of the Yoneda lemma.
+    <fr:link
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "The [language] is sorted…. We inductively define terms and their sorts:
-            <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
-                </fr:li>
-                <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
-                </fr:li>
-                <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
-                </fr:li></fr:ul>"
-            <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
+type="local"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    However, unlike <fr:link
+type="local"
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
+    so the natural transformation squares are well-defined.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
-            these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
-        </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Ignoring the natural numbers type and the idiosyncratic list type of
-        <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
-        elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
-        predicate logic" aka "topos logic" aka "higher-order intuitionistic
-        logic"</fr:link>, although certain inter-definable choices can be
-        made.
-    </fr:p><fr:p>
-        There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
-    </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
-<fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    While reading <fr:link
-href="von-neumann-1925.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
-    "that's… the point"-- "realization" that first order logic is the
-    meta-language of axiomatic mathematics, just as (it seems) logical
-    frameworks are the meta-languages of (typically λ-calculus based)
-    programming languages. Well not literally, but the discrepancy is
-    non-substantial.
-</fr:p><fr:p>
-    That is, (classical) FoL is often the setting in which axiomatic
-    mathematical theories are formalized. Classic (no pun intended) examples
-    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
-    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
-</fr:p><fr:p>
-    "definite property"
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
+        <fr:link
+type="local"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
+type="local"
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
+type="local"
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
+type="local"
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
     </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
+        This proof is rather low tech and is lifted straight from <fr:link
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
     </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
+        The forward directions are trivial.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
+        Then <fr:link
+type="local"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
+        true of an adjunction.
         <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
 type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
 type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+type="local"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
+                    <fr:embedded-tex
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+type="local"
+href="subb-0006.xml"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>428</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>472</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
-    </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
-        <fr:ul><fr:li>
-                completness theorems,
-            </fr:li>
-            <fr:li>
-                the solvability of higher type functional equations, and
-            </fr:li>
-            <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Fibered Right Adjoints are automatically fibered?
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
-    Do you only actually need functors? Zeilberger's <fr:link
-href="millies-zeilberger-2015.pdf"
-type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
-    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
 </fr:p><fr:p>
-    Talked to Max about this on 5-9.
-    They really just give to different notions of "logic over syntax"--
-    a hoare logic, and a implicational (eg FO) logic.
-    I suppose one'd want to study a bit the relationship between the two.
-    <fr:link
-href="millies-zeilberger-2015.xml"
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>478</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
-numbered="true"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>439</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Here is Blackburn et all's account:
 </fr:p><fr:p>
     The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "The basic case":
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
         </fr:li>
         <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
         <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
 
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>.
     A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
     Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
-</fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/hinman-2005.xml b/_site/forest/hinman-2005.xml
index e09fa2b..4848dd0 100644
--- a/_site/forest/hinman-2005.xml
+++ b/_site/forest/hinman-2005.xml
@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>449</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>553</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
     course text(s). I haven't used it much for reading either, owing to the
     large amount of new notation one has to refresh every time you open the
     book, but it's a solid reference book.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/index.xml b/_site/forest/index.xml
index 1017d54..d1d9690 100644
--- a/_site/forest/index.xml
+++ b/_site/forest/index.xml
@@ -1,93 +1,87 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="true"><fr:frontmatter><fr:anchor>803</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:contributor><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:contributor></fr:authors><fr:meta
+root="true"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:meta
 name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>277</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>219</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     I am a master's student in Computer Science and Engineering at the
     University of Michigan.
     I'm interested in logic and related areas of computer science, particularly
     programming languages.
 </fr:p><fr:p>
     This is my <fr:link
-href="https://www.jonmsterling.com/jms-005P.xml"
-type="external">forest</fr:link>.
+type="external"
+href="https://www.jonmsterling.com/jms-005P.xml">forest</fr:link>.
 </fr:p><fr:ul><fr:li>
                 email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li>
+hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble><![CDATA[]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[hejohns@umich.edu]]></fr:embedded-tex-body></fr:embedded-tex></fr:li>
     <fr:li>
         photo:
         <fr:p><html:img
 xmlns:html="http://www.w3.org/1999/xhtml"
 src="hejohns.jpeg"
 width="150"
-height="150"></html:img></fr:p></fr:li>
+height="150" /></fr:p></fr:li>
     <fr:li>
         cv: <fr:link
-href="cv.pdf"
-type="external">cv.pdf</fr:link> (TODO)
-    </fr:li></fr:ul></fr:mainmatter></fr:tree>
+type="external"
+href="cv.pdf">cv.pdf</fr:link> (TODO)
+    </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree>
     <fr:p>
         For more information about me, see <fr:link
-href="hejohns.xml"
 type="local"
-addr="hejohns">Johnson He</fr:link>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link>.
     </fr:p>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="true"
-show-heading="false"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>357</fr:anchor></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>311</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>223</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Some basic questions, whose "elevator pitch" answers ought to roll off
         the tongue:
         <fr:ul><fr:li>
@@ -108,9 +102,10 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
                 Is it the same thing as "topos logic"?
             </fr:li></fr:ul>
         Following <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
         Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
@@ -137,37 +132,41 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sket
         complete the sentence".
         A truly FAQ-style answer compendium will have to come later-- after I've
         actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -181,65 +180,75 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -255,107 +264,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -370,148 +402,196 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -532,248 +612,377 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -781,7 +990,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -789,540 +999,749 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
-    </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
+    </fr:p><fr:p>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>270</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Mentioned in passing as <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
         <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
     This is conventionally left implicit, but we remark to clarify what we mean
     by the family of natural isomorphisms being natural.
 </fr:p><fr:p>
     NOTE: I couldn't get よ to render in the latex (diagrams).
     I'm not sure if it's a lualatex/dvi or font issue.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -1330,376 +1749,413 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
     so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="true"
-show-heading="false"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>409</fr:anchor></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>360</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
     course text(s). I haven't used it much for reading either, owing to the
     large amount of new notation one has to refresh every time you open the
     book, but it's a solid reference book.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>362</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
-    title, and Makkai completeness.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>364</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    "Bibliographical Note: This Dover edition, first published in 2006, is a
-    slightly corrected unabridged republication of the revised (second) edition
-    of the work originally published in 1984 by Elsevier Science Publishers
-    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
-    "Studies in Logic and the Foundations of Mathematics." A new Preface and
-    updated Bibliography have been specially prepared for this reprint."
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="mclarty-1990a.pdf"
-type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -1729,8 +2185,11 @@ root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:ta
         and transformation of the thing. Doing this for "set theory" requires
         taking account of the experience that the main pairs of opposing
         tendencies in mathematics take the form of adjoint functors, and frees
-        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
-        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        us of the mathematically irrelevant traces (<fr:tex
+display="inline"><![CDATA[∈]]></fr:tex>) left behind by the
+        process of accumulating (<fr:tex
+display="inline"><![CDATA[∪]]></fr:tex>) the power set (<fr:tex
+display="inline"><![CDATA[\mathcal {P}]]></fr:tex>) at each stage of a
         metaphysical "construction". Further, experience with sheaves,
         permutation representations, algebraic spaces, etc., shows that a "set
         theory" for geometry should apply not only to abstract sets divorced
@@ -1738,52 +2197,41 @@ root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:ta
         sets which do in fact develop along such parameters. For such sets,
         usually logic is "intuitionistic" (in its formal properties)
         usually the axiom of choice is false, and usually a set is not
-        determined by its points defined over <fr:tex>1</fr:tex> only.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+        determined by its points defined over <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> only.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     I have not read this, and have no idea what "contradictions" he's referring to.
     <fr:link
-href="lawvere-1970.pdf"
-type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="lawvere-1970.pdf">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -1794,188 +2242,170 @@ root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Abstract</fr:ta
         remarks on a categorical view of the history of set theory, including a
         false history plausible from that point of view that would make it helpful
         to introduce toposes as a generalization from set theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="mclarty-1990b.pdf"
-type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="mclarty-1990b.pdf">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
-href="ronnie-chen.xml"
 type="local"
-addr="ronnie-chen">Ronnie Chen</fr:link>,
+href="ronnie-chen.xml"
+addr="ronnie-chen"
+title="Ronnie Chen">Ronnie Chen</fr:link>,
     where the viewpoint of "generalized algebraic semantics" is developed.
 </fr:p><fr:p><fr:link
-href="marquis-reyes-2012.pdf"
-type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="marquis-reyes-2012.pdf">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        This is the written version of a talk given at the C.I.R.M. Workshop on
-        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
-        is to argue that there is a close connection between the theory of
-        computation and the geometric side of topos theory. We begin with a brief
-        outline of the history and basic concepts of topos theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Prof. Blass essentially described this background over the course of
-        the logic seminar last semester (as comments at various meetings,
-        culminating with a full talk), but this is a nice written account of
-        the history of topos theory, the relation between Grothendieck and
-        Elementary topoi, and the relation between geometric and logical
-        morphisms.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="blass-1988.pdf"
-type="external">blass-1988.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    "Bibliographical Note: This Dover edition, first published in 2006, is a
+    slightly corrected unabridged republication of the revised (second) edition
+    of the work originally published in 1984 by Elsevier Science Publishers
+    B.V., Amsterdam, The Netherlands, as Volume 98 in the North-Holland Series,
+    "Studies in Logic and the Foundations of Mathematics." A new Preface and
+    updated Bibliography have been specially prepared for this reprint."
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+type="local"
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">Equality Between Functionals</fr:link> and <fr:link
+type="local"
+href="statman-1985.xml"
+addr="statman-1985"
+title="Equality Between Functionals Revisited">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+type="external"
+href="https://proofassistants.stackexchange.com/a/1177">stack overflow</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    One of the early NBG papers. Maybe "the" NBG paper.
-    My notes on this are currently in <fr:link
-href="quine-1960.xml"
-type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
-</fr:p><fr:p>
-    English translation in <fr:link
-href="van-heijenoort-1967.xml"
-type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The λ-calculus, in both its typed and untyped forms, has primarily been
+        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
+        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
+        It seems fair to say that this aspect of the λ-calculus is currently in an
+        embryonic state of development, awaiting further conceptual advances.
+    </fr:p><fr:p>
+        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
+        readily understood. This is its connection with the full classical finite
+        type structure over ω, (i.e., with the functionals of finite type). In an
+        obvious way, each closed term in the typed λ-calculus defines a functional
+        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
+        some closed term in the typed x-calculus.
+    </fr:p><fr:p>
+        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
+        with or without types, have been considered. The motivation for introducing
+        these definitions has primarily been to analyze the notion of the
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex
+display="inline"><![CDATA[⊢ s = t]]></fr:tex> defined
+        in the text, is equivalent to one of these definitions of convertibility.
+        It is easy to see that any two convertible terms define the same functional
+        of finite type. We show here that any two non- convertible terms define
+        different functionals of finite type.[This result was obtained in 1970.]
+        This, coupled with the known decidability of convertibility, tells us that
+        "equality between simple functionals is recursive."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Square brackets denote author's footnotes.
+    <fr:link
+type="external"
+href="friedman-1975.pdf">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
-    this list. Whatever else you might think about Quine, he was a terrific
-    explainer, and this is the best introduction to the subject I know."
-</fr:p><fr:p>
-    (Yes, that 50 should be a 60(b).)
-    Here is the followup I wrote:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        In this note we shall try to place Friedman's remarkable little paper
+        (<fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">3</fr:link>) in the context of what we know today about the model
+        theory of the typed λ-calculus. In order to do this, it is appropriate to
+        survey recent work in this area, in so far as it touches on issues raised
+        by Friedman. We don't intend a general survey; much will be left out. In
+        particular, we shall omit discussion of unification, fragments of L.C.F.,
+        and polymorphic types, which are areas of interest to the author, and the
+        specialist will surely find other omissions. However, we will try to show
+        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
+        model theory of the typed λ-calculus.
     </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        The plan of this note is the following. We shall begin by a brief, informal
+        introduction to the subject. The reader is then advised to read Friedman's
+        paper. It is, after all, quite accessible and easy to read. We shall then
+        proceed to consider the three major issues touched on in "Equality between
+        functionals" as we view them today. These issues are
+        <fr:ul><fr:li>
+                completness theorems,
+            </fr:li>
+            <fr:li>
+                the solvability of higher type functional equations, and
+            </fr:li>
+            <fr:li>
+                logical relations.
+            </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="statman-1985.pdf">statman-1985.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -1996,14 +2426,14 @@ root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Abstract</fr:ta
         stuck.  To achieve self-interpretation and show Jones-optimality, we
         relate this exemplar of higher-rank and higher-kind polymorphism to
         plugging a term into a context of let-polymorphic bindings.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -2025,21 +2455,116 @@ root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Notes</fr:taxon
         that it is the free syntax generated by the encoded rules/initial
         algebra/inductive dataype.
         But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex
+display="inline"><![CDATA[∀M]]></fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -2053,14 +2578,14 @@ root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:ta
         computability</fr:strong>, logical relations are manipulated as if they were sets,
         making the essence of many complex logical relations arguments
         accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -2073,327 +2598,208 @@ root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</f
         and I just don't have some context, or it's actually presented almost
         mystically.
         The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+href="carette-kiselyov-shan-2007.xml"
+addr="carette-kiselyov-shan-2007"
+title="Finally Tagless, Partially Evaluated">finally tagless encodings</fr:link>
         and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
+type="external"
+href="hoffman-1999">a justification for hoas</fr:link> doesn't seem super
         on-the-point, but if it's mentioned it must be for good reason…
     </fr:p><fr:p>
         Again, quotes are air-quotes, not actual quotes.
         <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="sterling-2022.pdf">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Mainstream orthodoxy holds that set theory provides a foundation for
-        contemporary pure mathematics. Critics of this view argue that category
-        theory, or more recently univalent foundations, is better suited to this
-        role. Some observers of this controversy suggest that it might be resolved
-        by a better understanding of what a foundation is. Despite considerable
-        sympathy to this line of thought, I’m skeptical of the unspoken assumption
-        that there’s an underlying concept of a ‘foundation’ up for analysis, that
-        this analysis would properly guide our assessment of the various
-        candidates. In contrast, it seems to me that the considerations the
-        combatants offer against opponents and for their preferred candidates, as
-        well as the roles each candidate actually or potentially succeeds in
-        playing, reveal quite a number of different jobs that mathematicians want
-        done. What matters is these jobs we want our theories to do and how well
-        they do them. Whether any or all of them, jobs or theories, deserves to be
-        called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    A nice article that can be briefly summarized as
-    <fr:ul><fr:li>
-            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
-            system, whatever that means
-        </fr:li>
-        <fr:li>
-            Classical set theory is perfectly fine as a foundational system in
-            the traditionally (ie ZFC) conceived sense
-        </fr:li>
-        <fr:li>
-            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
-            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
-            bleeding edge").
-        </fr:li></fr:ul></fr:p><fr:p>
-    The contrast between the minimalistic ZFC and potentially enormous "user
-    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
-    natural deduction. Indeed, it's like how Ronnie calls set theory "the
-    machine/assembly language of math" (kind of paraphrase), and Max calls SK
-    combinators "the machine/assembly language of λ-calculus".
-</fr:p><fr:p>
-    Natural deduction is certainly more complicated than SK/Hilbert proof
-    systems, but has nice usability properties that are worth study and
-    implementation.
-    The general mathematical trend seems to be that of starting with a spartan
-    formal system, then gradually making it more user friendly to accomodate
-    every increasingly complicated mathematics, where the minimalistic system
-    starts to leave one wanting.
-</fr:p><fr:p>
-    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
-    redundant to mention other proof assistants eg Cubical Agda.
-</fr:p><fr:p>
-    I've used quotes here as air quotes, not actual quotes of Maddy.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
     <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="blass-2003.pdf">blass-2003.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>394</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The λ-calculus, in both its typed and untyped forms, has primarily been
-        regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
-        <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
-        It seems fair to say that this aspect of the λ-calculus is currently in an
-        embryonic state of development, awaiting further conceptual advances.
-    </fr:p><fr:p>
-        There is, however, another aspect of the <fr:em>typed</fr:em> λ-calculus, which is
-        readily understood. This is its connection with the full classical finite
-        type structure over ω, (i.e., with the functionals of finite type). In an
-        obvious way, each closed term in the typed λ-calculus defines a functional
-        of finite type over ω.  Call a functional <fr:em>simple</fr:em> if it is given by
-        some closed term in the typed x-calculus.
-    </fr:p><fr:p>
-        Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
-        with or without types, have been considered. The motivation for introducing
-        these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
-        in the text, is equivalent to one of these definitions of convertibility.
-        It is easy to see that any two convertible terms define the same functional
-        of finite type. We show here that any two non- convertible terms define
-        different functionals of finite type.[This result was obtained in 1970.]
-        This, coupled with the known decidability of convertibility, tells us that
-        "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    Square brackets denote author's footnotes.
-    <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        This is the written version of a talk given at the C.I.R.M. Workshop on
+        Logic in Computer Science at Marseille, France in June, 1988. Its purpose
+        is to argue that there is a close connection between the theory of
+        computation and the geometric side of topos theory. We begin with a brief
+        outline of the history and basic concepts of topos theory.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        In this note we shall try to place Friedman's remarkable little paper
-        (<fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
-        theory of the typed λ-calculus. In order to do this, it is appropriate to
-        survey recent work in this area, in so far as it touches on issues raised
-        by Friedman. We don't intend a general survey; much will be left out. In
-        particular, we shall omit discussion of unification, fragments of L.C.F.,
-        and polymorphic types, which are areas of interest to the author, and the
-        specialist will surely find other omissions. However, we will try to show
-        the reader how Friedman's paper <fr:em>lays the foundation</fr:em> for the general
-        model theory of the typed λ-calculus.
-    </fr:p><fr:p>
-        The plan of this note is the following. We shall begin by a brief, informal
-        introduction to the subject. The reader is then advised to read Friedman's
-        paper. It is, after all, quite accessible and easy to read. We shall then
-        proceed to consider the three major issues touched on in "Equality between
-        functionals" as we view them today. These issues are
-        <fr:ul><fr:li>
-                completness theorems,
-            </fr:li>
-            <fr:li>
-                the solvability of higher type functional equations, and
-            </fr:li>
-            <fr:li>
-                logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Prof. Blass essentially described this background over the course of
+        the logic seminar last semester (as comments at various meetings,
+        culminating with a full talk), but this is a nice written account of
+        the history of topos theory, the relation between Grothendieck and
+        Elementary topoi, and the relation between geometric and logical
+        morphisms.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="blass-1988.pdf">blass-1988.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    One of the early NBG papers. Maybe "the" NBG paper.
+    My notes on this are currently in <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+href="quine-1960.xml"
+addr="quine-1960"
+title="Variables Explained Away">Variables Explained Away</fr:link>.
+</fr:p><fr:p>
+    English translation in <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The opening sentence is what <fr:link
-href="zeilberger"
-type="external">zeilberger</fr:link> quotes on his work on
-        polarity.
-    </fr:p><fr:p>
-        What's the relationship between harmony-stability and Pfenning's local
-        soundness and completeness?
-        They're not literally the same thing, but they're similar.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="van-heijenoort-1967.xml"
+addr="van-heijenoort-1967"
+title="From Frege To Gödel">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A+. I should get a second copy to gift whenever needed.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The standard reading of type theory through the lens of category theory
-        is based on the idea of viewing a type system as a category of
-        well-typed terms. We propose a basic revision of this reading: rather
-        than interpreting type systems as categories, we describe them as
-        functors from a category of typing derivations to a category of
-        underlying terms. Then, turning this around, we explain how in fact any
-        functor gives rise to a generalized type system, with an abstract
-        notion of typing judgment, typing derivations and typing rules. This
-        leads to a purely categorical reformulation of various natural classes
-        of type systems as natural classes of functors. The main purpose of
-        this paper is to describe the general framework (which can also be seen
-        as providing a categorical analysis of refinement types), and to
-        present a few applications. As a larger case study, we revisit
-        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
-        paper’s main results may be reconstructed along these lines.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
+    this list. Whatever else you might think about Quine, he was a terrific
+    explainer, and this is the best introduction to the subject I know."
+</fr:p><fr:p>
+    (Yes, that 50 should be a 60(b).)
+    Here is the followup I wrote:
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
-        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
-        functor into) the syntax, typically trivially constructed as a pullback
-        (it's hard in general to concoct models with structure preserving maps
-        into the syntax).
-        This is so we can take advantage of the universal property of the
-        syntax (its initiality), as we always do (eg classical soundness of
-        Prop logic).
-        The initiality lets us lift any syntactic term into the model, and talk
-        about it in the model. The projection of the model then let's us
-        actually say something about the syntax itself.
-        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
-        to let us say something interesting/talk about what we care about.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Thanks for sharing this nice little note, which sketches a combinatory
+        (classical) first-order logic.
     </fr:p><fr:p>
-        So (classical) parametricity (a la Reynolds '83) is just this.
-        He constructs a set-theoretic model of system F (all models in the
-        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
-        In terms of data abstraction/automata, the point is to construct a
-        bisimulation between the two models, so we can say that two types are
-        "the same"-- every step in the program preserves the relation in
-        lock-step.
-        This is entirely analogous to canonicity, where we need to actually
-        reason about the whole language preserving the type, not just intro and
-        elims for that type, for a stronger inductive hypothesis.
+        I was surprised by the mention of von Neumann 1925 in this snippet:
     </fr:p><fr:p>
-        This way, we can show that two different semantics and implementations
-        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
-        typically relate a naive implementation and a optimized implementation.
-        That is, the reference and optimized implementations are
-        bisimilar/extensionally equivalent.
+        "In a way our operators are reminiscent also of the axioms of class
+        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
+        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
+        dissimilarity of purpose." (Footnote 1, p. 346)
     </fr:p><fr:p>
-        I'm still not super clear on why parametricity fundamentally is tied
-        with this notion of data hiding.  I guess it's that to give an account
-        of "operating on a type, just knowing it's a type", one has to give a
-        notion of "just knowing it's a type".  For this, Reynolds defines a
-        notion of "respecting interfaces" w/ relations between models, and that
-        polymophic terms respect any interface?
+        From what I gather, the reminiscence with von Neumann 1925 (I assume
+        Quine was principally referring to the "III. Logical Construction
+        Axioms") is that the purposes (of Quine's predicate combinators and von
+        Neumann's axioms of existence of II-objects) are actually quite
+        similar.
     </fr:p><fr:p>
-        That last question is that the "constructing a bisimulation between two
-        models" view seems to make sense, but as a phrasing of parametricity,
-        it should really be that "for any two models, we can always construct
-        (using a LR) a bisimulation between them".
-        Assuming it's actually true eg you can't construct the model if you say
-        that the empty intlist is related to a nonempty one.
+        It seems that the system of von Neumann 1925 isn't couched in any
+        particular formal system (although it's maybe implicitly first-order,
+        and apparently translated into such by Skolem 1938), thus von Neumann
+        wants to recover from any definite property ("logical condition") on
+        I-objects, the (existence of a) class (II-object) of those I-objects.
+        Wikipedia calls this the <fr:link
+type="external"
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem">class existence theorem</fr:link>.
     </fr:p><fr:p>
-        And do we get anything interesting if one of the models is itself the syntax?
-        Is this what parametricity is?
-        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
-        What about the syntax as a "closed terms" henkin model?
+        (More generally, the "reduction theorem" in von Neumann 1925 says that
+        every expression of I-objects (definable class-function) is indeed a
+        II-object (class-function). This restricts to logical conditions and
+        classes.)
     </fr:p><fr:p>
-        Maybe I'm finally close to being able to understand what a Kripke
-        logical relation is.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        To do this economically, the existence of a few basic
+        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
+        conditions are put in "normal form" by eliminating variables with the
+        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
+        working in an explicit logic, would be to just stipulate a schema of
+        class comprehension (also mentioned in the Wikipedia article), as
+        included in the back of Chang and Keisler.
+    </fr:p><fr:p>
+        Actually, while there are explicit class existence axioms for absorbing
+        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
+        they'd follow with the other axioms, but I suppose they must. The
+        Wikipedia article has a write-up starting from more user-friendly
+        axioms.
+    </fr:p><fr:p>
+        In any case, I hadn't realized that the historic move to make precise
+        the notion of 'definite property' as 'first-order definable' was
+        nontrivial, and found this connection with Quine's note interesting,
+    </fr:p><fr:p>
+        Johnson He
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:ul><fr:li><fr:link
+type="external"
+href="quine-1960.pdf">quine-1960.pdf</fr:link></fr:li>
+    <fr:li><fr:link
+type="local"
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -2402,221 +2808,298 @@ root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:ta
         only on objects and the usual hom adjointness isomorphism implied the
         effect of the functors on homs was uniquely determined. Writing this down
         properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Mainstream orthodoxy holds that set theory provides a foundation for
+        contemporary pure mathematics. Critics of this view argue that category
+        theory, or more recently univalent foundations, is better suited to this
+        role. Some observers of this controversy suggest that it might be resolved
+        by a better understanding of what a foundation is. Despite considerable
+        sympathy to this line of thought, I’m skeptical of the unspoken assumption
+        that there’s an underlying concept of a ‘foundation’ up for analysis, that
+        this analysis would properly guide our assessment of the various
+        candidates. In contrast, it seems to me that the considerations the
+        combatants offer against opponents and for their preferred candidates, as
+        well as the roles each candidate actually or potentially succeeds in
+        playing, reveal quite a number of different jobs that mathematicians want
+        done. What matters is these jobs we want our theories to do and how well
+        they do them. Whether any or all of them, jobs or theories, deserves to be
+        called ‘foundational’ is really beside the point.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    A nice article that can be briefly summarized as
+    <fr:ul><fr:li>
+            There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
+            system, whatever that means
+        </fr:li>
+        <fr:li>
+            Classical set theory is perfectly fine as a foundational system in
+            the traditionally (ie ZFC) conceived sense
+        </fr:li>
+        <fr:li>
+            There is promise for the new 21ˢᵗ century desiderata of <fr:em>proof
+            checking</fr:em> (recalling Voevodsky's concerns with correctness "at the
+            bleeding edge").
+        </fr:li></fr:ul></fr:p><fr:p>
+    The contrast between the minimalistic ZFC and potentially enormous "user
+    friendly" univalent type theories is reminiscent of Hilbert style proofs vs
+    natural deduction. Indeed, it's like how Ronnie calls set theory "the
+    machine/assembly language of math" (kind of paraphrase), and Max calls SK
+    combinators "the machine/assembly language of λ-calculus".
+</fr:p><fr:p>
+    Natural deduction is certainly more complicated than SK/Hilbert proof
+    systems, but has nice usability properties that are worth study and
+    implementation.
+    The general mathematical trend seems to be that of starting with a spartan
+    formal system, then gradually making it more user friendly to accomodate
+    every increasingly complicated mathematics, where the minimalistic system
+    starts to leave one wanting.
+</fr:p><fr:p>
+    Maddy only mentions Coq, but being 2019 it would've been nice is somewhat
+    redundant to mention other proof assistants eg Cubical Agda.
+</fr:p><fr:p>
+    I've used quotes here as air quotes, not actual quotes of Maddy.
+    <fr:link
+type="external"
+href="maddy-2019.pdf">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>dummett-91</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The opening sentence is what <fr:link
+type="external"
+href="zeilberger">zeilberger</fr:link> quotes on his work on
+        polarity.
+    </fr:p><fr:p>
+        What's the relationship between harmony-stability and Pfenning's local
+        soundness and completeness?
+        They're not literally the same thing, but they're similar.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        TODO: not a good selection
-    </fr:p><fr:ul><fr:li><fr:link
-href="https://github.com/eecs490/gradescope-utils"
-type="external">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Contributing to <fr:link
+type="external"
+href="https://github.com/maxsnew/cubical-categorical-logic">Cubical Categorical Logic</fr:link>, "cubical agda formalizations of different approaches to categorical semantics of type theories"
         </fr:li>
         <fr:li><fr:link
-href="https://github.com/hejohns/.rc/tree/master/home/bin"
-type="external">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
-href="https://mplse-reading-group.github.io/"
-type="external">https://mplse-reading-group.github.io/</fr:link></fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+type="external"
+href="https://mplse-reading-group.github.io/">https://mplse-reading-group.github.io/</fr:link></fr:li>
         <fr:li><fr:link
-href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler"
-type="external">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://github.com/hejohns/the-second-pass-of-the-portable-c-compiler">The Second Pass of the Portable C Compiler</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
-href="../posts.html"
-type="external">Jekyll Blog</fr:link></fr:mainmatter></fr:tree>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
+type="external"
+href="../posts.html">Jekyll Blog</fr:link></fr:mainmatter><fr:backmatter /></fr:tree>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="true"
-show-heading="false"
-show-metadata="false"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>443</fr:anchor></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html"
-type="external">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
-href="stack-overflow/every-topos-is-a-regular-category.html"
-type="external">stack-overflow/every-topos-is-a-regular-category.html</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p /></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
-    title, and Makkai completeness.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
+type="external"
+href="stack-overflow/every-topos-is-a-regular-category.html">stack-overflow/every-topos-is-a-regular-category.html</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
+    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
+    up time and time again, and makes an appearance in Tom de Jong's <fr:link
+type="external"
+href="https://github.com/tomdjong/MGS-categorical-realizability">notes on
+    realizability</fr:link>,
+    according to Loris it's relevant to Markov Decision Processes, and we
+    develop the view that it's a logician's view of fibrations.
+</fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        The unity of opposites in the title is essentially that between logic
-        and geometry, and there are compelling reasons for maintaining that
-        geometry is the leading aspect. At the same lime, in the present joint
-        work with Myles Tierney there are important influences in the other
-        direction: a Grothendieck "topology" appears most naturally as a
-        modal operator, of the nature "it is locally the case that", the
-        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
-        apply to families of geometrical objects rather than to propositional
-        functions, and an important technique is to lift constructions first
-        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
-        topos. We first sum up the principal contradictions of the
-        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
-        adjoint functors, significantly generalizing the theory to free it of
-        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
-        enabling one to claim that in a sense logic is a special case of
-        geometry). The method thus developing is then applied to intrinsically
-        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
-        the independence of the continuum hypothesis free of any use of
-        transfinite induction. The second application of the method outlined
-        here is an intrinsic geometric construction of the Chevalley-Hakim
-        global spectrum of a ringed topos free of any choice of a "site of
-        definition".
-    </fr:p><fr:p>
-        When the main contradictions of a thing have been found,
-        the scientific procedure is to summarize them in slogans which one then
-        constantly uses as an ideological weapon for the further development
-        and transformation of the thing. Doing this for "set theory" requires
-        taking account of the experience that the main pairs of opposing
-        tendencies in mathematics take the form of adjoint functors, and frees
-        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
-        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
-        metaphysical "construction". Further, experience with sheaves,
-        permutation representations, algebraic spaces, etc., shows that a "set
-        theory" for geometry should apply not only to abstract sets divorced
-        from time, space, ring of definition, etc., but also to more general
-        sets which do in fact develop along such parameters. For such sets,
-        usually logic is "intuitionistic" (in its formal properties)
-        usually the axiom of choice is false, and usually a set is not
-        determined by its points defined over <fr:tex>1</fr:tex> only.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    I have not read this, and have no idea what "contradictions" he's referring to.
-    <fr:link
-href="lawvere-1970.pdf"
-type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Here is Blackburn et all's account:
+</fr:p><fr:p>
+    The concept first presented is that of <fr:tree
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "The basic case":
+</fr:p><fr:p>
+    A relation <fr:tex
+display="inline"><![CDATA[Z]]></fr:tex> (to not be confused with the accessibility relations <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> for
+    each Kripke frame) between two Kripke frames <fr:tex
+display="inline"><![CDATA[M]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[M]]></fr:tex>' is a *bisimulation* iff
+    <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[w Z w']]></fr:tex> ⟹ (<fr:tex
+display="inline"><![CDATA[w]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[w]]></fr:tex>' are elementarily equivalent)
+        </fr:li>
+        <fr:li>
+            (<fr:em>forth</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v']]></fr:tex></fr:li>
+        <fr:li>
+            (<fr:em>backwards</fr:em>) <fr:tex
+display="inline"><![CDATA[w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v']]></fr:tex></fr:li></fr:ul></fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>370</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
-        resource consciousness that makes sense in classical predicate logic.
-        We characterize the propositional formulas all of whose first-order
-        instances are simply Herbrand valid. The characterization turns out to
-        coincide with a known characterization of game semantical validity for
-        multiplicative formulas.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
-    This is currently a "would like to read, but no urgency" paper, because
-    Blass + Herbrand's theorem.
-    <fr:link
-href="blass-2003.pdf"
-type="external">blass-2003.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
+
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>.
+    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
 show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    I have not read this. This might be a good source, aside from <fr:link
-href="ronnie-chen.xml"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="ronnie-chen">Ronnie Chen</fr:link>,
-    where the viewpoint of "generalized algebraic semantics" is developed.
-</fr:p><fr:p><fr:link
-href="marquis-reyes-2012.pdf"
-type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Clearly the truth of <fr:tex
+display="inline"><![CDATA[∧,∨,◇]]></fr:tex>-formulas are preserved under simulations.
+    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Some basic questions, whose "elevator pitch" answers ought to roll off
         the tongue:
         <fr:ul><fr:li>
@@ -2637,9 +3120,10 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
                 Is it the same thing as "topos logic"?
             </fr:li></fr:ul>
         Following <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
         Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
@@ -2666,37 +3150,41 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sket
         complete the sentence".
         A truly FAQ-style answer compendium will have to come later-- after I've
         actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -2710,65 +3198,75 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -2784,107 +3282,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -2899,148 +3420,196 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -3061,248 +3630,377 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -3310,7 +4008,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -3318,262 +4017,346 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
-    </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
+    </fr:p><fr:p>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    While reading <fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
-    "that's… the point"-- "realization" that first order logic is the
-    meta-language of axiomatic mathematics, just as (it seems) logical
-    frameworks are the meta-languages of (typically λ-calculus based)
-    programming languages. Well not literally, but the discrepancy is
-    non-substantial.
-</fr:p><fr:p>
-    That is, (classical) FoL is often the setting in which axiomatic
-    mathematical theories are formalized. Classic (no pun intended) examples
-    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
-    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
-</fr:p><fr:p>
-    "definite property"
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
+    title, and Makkai completeness.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
-        properties of logics, type theories, and programming languages:
-        canonicity, decidability, conservativity, computational adequacy, and
-        more. Logical relations combine pure syntax with non-syntactic objects
-        that are parameretized in syntax in a somewhat complex way; the
-        sophistication of possible parameterizations makes logical relations a
-        tool that is primarily accessible to specialists. In the spirit of
-        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
-        relations based on synthetic Tait computability, an internal language
-        for categories of logical relations. In <fr:strong>synthetic Tait
-        computability</fr:strong>, logical relations are manipulated as if they were sets,
-        making the essence of many complex logical relations arguments
-        accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The unity of opposites in the title is essentially that between logic
+        and geometry, and there are compelling reasons for maintaining that
+        geometry is the leading aspect. At the same lime, in the present joint
+        work with Myles Tierney there are important influences in the other
+        direction: a Grothendieck "topology" appears most naturally as a
+        modal operator, of the nature "it is locally the case that", the
+        usual logical operators such as ∀, ∃, ⇒ have natural analogues which
+        apply to families of geometrical objects rather than to propositional
+        functions, and an important technique is to lift constructions first
+        understood for "the" category <fr:strong>S</fr:strong> of abstract sets to an arbitrary
+        topos. We first sum up the principal contradictions of the
+        Grothendieck-Giraud-Verdier theory of topos in terms of four or five
+        adjoint functors, significantly generalizing the theory to free it of
+        reliance on an <fr:em>external</fr:em> notion of infinite limit (in particular
+        enabling one to claim that in a sense logic is a special case of
+        geometry). The method thus developing is then applied to intrinsically
+        define the concept of Boolean-valued model for <fr:strong>S</fr:strong> (<fr:strong>BVM/S</fr:strong>) and to prove
+        the independence of the continuum hypothesis free of any use of
+        transfinite induction. The second application of the method outlined
+        here is an intrinsic geometric construction of the Chevalley-Hakim
+        global spectrum of a ringed topos free of any choice of a "site of
+        definition".
+    </fr:p><fr:p>
+        When the main contradictions of a thing have been found,
+        the scientific procedure is to summarize them in slogans which one then
+        constantly uses as an ideological weapon for the further development
+        and transformation of the thing. Doing this for "set theory" requires
+        taking account of the experience that the main pairs of opposing
+        tendencies in mathematics take the form of adjoint functors, and frees
+        us of the mathematically irrelevant traces (<fr:tex
+display="inline"><![CDATA[∈]]></fr:tex>) left behind by the
+        process of accumulating (<fr:tex
+display="inline"><![CDATA[∪]]></fr:tex>) the power set (<fr:tex
+display="inline"><![CDATA[\mathcal {P}]]></fr:tex>) at each stage of a
+        metaphysical "construction". Further, experience with sheaves,
+        permutation representations, algebraic spaces, etc., shows that a "set
+        theory" for geometry should apply not only to abstract sets divorced
+        from time, space, ring of definition, etc., but also to more general
+        sets which do in fact develop along such parameters. For such sets,
+        usually logic is "intuitionistic" (in its formal properties)
+        usually the axiom of choice is false, and usually a set is not
+        determined by its points defined over <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> only.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    I have not read this, and have no idea what "contradictions" he's referring to.
+    <fr:link
+type="external"
+href="lawvere-1970.pdf">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
-        idea of what's going on starting at p. 9.
-    </fr:p><fr:p>
-        From what I can tell, the idea is to work in the internal language of some
-        topos-- really the internal "agda-like dependent type theory"-- to
-        construct the logical relations category, instead of doing something like
-        (analytic) gluing.
-        How this makes anything easier is beyond me at the moment.
-        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
-        and I just don't have some context, or it's actually presented almost
-        mystically.
-        The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    I have not read this. This might be a good source, aside from <fr:link
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
-        and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
-        on-the-point, but if it's mentioned it must be for good reason…
-    </fr:p><fr:p>
-        Again, quotes are air-quotes, not actual quotes.
-        <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="ronnie-chen.xml"
+addr="ronnie-chen"
+title="Ronnie Chen">Ronnie Chen</fr:link>,
+    where the viewpoint of "generalized algebraic semantics" is developed.
+</fr:p><fr:p><fr:link
+type="external"
+href="marquis-reyes-2012.pdf">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
+type="local"
 href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">Equality Between Functionals</fr:link> and <fr:link
 type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
 href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+addr="statman-1985"
+title="Equality Between Functionals Revisited">Equality Between Functionals Revisited</fr:link>.
     Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://proofassistants.stackexchange.com/a/1177">stack overflow</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -3590,37 +4373,39 @@ root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:ta
         Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
         with or without types, have been considered. The motivation for introducing
         these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex
+display="inline"><![CDATA[⊢ s = t]]></fr:tex> defined
         in the text, is equivalent to one of these definitions of convertibility.
         It is easy to see that any two convertible terms define the same functional
         of finite type. We show here that any two non- convertible terms define
         different functionals of finite type.[This result was obtained in 1970.]
         This, coupled with the known decidability of convertibility, tells us that
         "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     Square brackets denote author's footnotes.
     <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="friedman-1975.pdf">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>428</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
-href="friedman-1975.xml"
 type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">3</fr:link>) in the context of what we know today about the model
         theory of the typed λ-calculus. In order to do this, it is appropriate to
         survey recent work in this area, in so far as it touches on issues raised
         by Friedman. We don't intend a general survey; much will be left out. In
@@ -3643,182 +4428,259 @@ addr="friedman-1975">3</fr:link>) in the context of what we know today about the
             </fr:li>
             <fr:li>
                 logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="statman-1985.pdf">statman-1985.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Fibered Right Adjoints are automatically fibered?
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="false"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="false"
-root="false"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
-    Do you only actually need functors? Zeilberger's <fr:link
-href="millies-zeilberger-2015.pdf"
-type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
-    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
-</fr:p><fr:p>
-    Talked to Max about this on 5-9.
-    They really just give to different notions of "logic over syntax"--
-    a hoare logic, and a implicational (eg FO) logic.
-    I suppose one'd want to study a bit the relationship between the two.
-    <fr:link
-href="millies-zeilberger-2015.xml"
-type="local"
-addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    Fibered Right Adjoints are automatically fibered?
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>This is a fundamental notion.</fr:p><fr:p>
-    I first saw this in modal logic a la Blackburn, de Rijke, Venema, but comes
-    up time and time again, and makes an appearance in Tom de Jong's <fr:link
-href="https://github.com/tomdjong/MGS-categorical-realizability"
-type="external">notes on
-    realizability</fr:link>,
-    according to Loris it's relevant to Markov Decision Processes, and we
-    develop the view that it's a logician's view of fibrations.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>439</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Here is Blackburn et all's account:
-</fr:p><fr:p>
-    The concept first presented is that of <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        The standard reading of type theory through the lens of category theory
+        is based on the idea of viewing a type system as a category of
+        well-typed terms. We propose a basic revision of this reading: rather
+        than interpreting type systems as categories, we describe them as
+        functors from a category of typing derivations to a category of
+        underlying terms. Then, turning this around, we explain how in fact any
+        functor gives rise to a generalized type system, with an abstract
+        notion of typing judgment, typing derivations and typing rules. This
+        leads to a purely categorical reformulation of various natural classes
+        of type systems as natural classes of functors. The main purpose of
+        this paper is to describe the general framework (which can also be seen
+        as providing a categorical analysis of refinement types), and to
+        present a few applications. As a larger case study, we revisit
+        Reynolds’ paper on “The Meaning of Types” (2000), showing how the
+        paper’s main results may be reconstructed along these lines.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex
+display="inline"><![CDATA[∀M]]></fr:tex> statements.
+        Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
+        functor into) the syntax, typically trivially constructed as a pullback
+        (it's hard in general to concoct models with structure preserving maps
+        into the syntax).
+        This is so we can take advantage of the universal property of the
+        syntax (its initiality), as we always do (eg classical soundness of
+        Prop logic).
+        The initiality lets us lift any syntactic term into the model, and talk
+        about it in the model. The projection of the model then let's us
+        actually say something about the syntax itself.
+        So a logical relations argument is <fr:em>just</fr:em> a clever choice of model,
+        to let us say something interesting/talk about what we care about.
+    </fr:p><fr:p>
+        So (classical) parametricity (a la Reynolds '83) is just this.
+        He constructs a set-theoretic model of system F (all models in the
+        paper are set theoretic) out of <fr:em>two</fr:em> standard models.
+        In terms of data abstraction/automata, the point is to construct a
+        bisimulation between the two models, so we can say that two types are
+        "the same"-- every step in the program preserves the relation in
+        lock-step.
+        This is entirely analogous to canonicity, where we need to actually
+        reason about the whole language preserving the type, not just intro and
+        elims for that type, for a stronger inductive hypothesis.
+    </fr:p><fr:p>
+        This way, we can show that two different semantics and implementations
+        for a <fr:code>intset</fr:code> type are in a precise sense, "the same".  We
+        typically relate a naive implementation and a optimized implementation.
+        That is, the reference and optimized implementations are
+        bisimilar/extensionally equivalent.
+    </fr:p><fr:p>
+        I'm still not super clear on why parametricity fundamentally is tied
+        with this notion of data hiding.  I guess it's that to give an account
+        of "operating on a type, just knowing it's a type", one has to give a
+        notion of "just knowing it's a type".  For this, Reynolds defines a
+        notion of "respecting interfaces" w/ relations between models, and that
+        polymophic terms respect any interface?
+    </fr:p><fr:p>
+        That last question is that the "constructing a bisimulation between two
+        models" view seems to make sense, but as a phrasing of parametricity,
+        it should really be that "for any two models, we can always construct
+        (using a LR) a bisimulation between them".
+        Assuming it's actually true eg you can't construct the model if you say
+        that the empty intlist is related to a nonempty one.
+    </fr:p><fr:p>
+        And do we get anything interesting if one of the models is itself the syntax?
+        Is this what parametricity is?
+        What if we put the same model on both sides? It should be trivial, thinking of it as a bisimulation, but I guess it just gives you a logical predicate.
+        What about the syntax as a "closed terms" henkin model?
+    </fr:p><fr:p>
+        Maybe I'm finally close to being able to understand what a Kripke
+        logical relation is.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
 expanded="false"
-root="false"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    While reading <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "The basic case":
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
+    "that's… the point"-- "realization" that first order logic is the
+    meta-language of axiomatic mathematics, just as (it seems) logical
+    frameworks are the meta-languages of (typically λ-calculus based)
+    programming languages. Well not literally, but the discrepancy is
+    non-substantial.
 </fr:p><fr:p>
-    A relation <fr:tex>Z</fr:tex> (to not be confused with the accessibility relations <fr:tex>R</fr:tex> for
-    each Kripke frame) between two Kripke frames <fr:tex>M</fr:tex> and <fr:tex>M</fr:tex>' is a *bisimulation* iff
-    <fr:ul><fr:li><fr:tex>w Z w'</fr:tex> ⟹ (<fr:tex>w</fr:tex> and <fr:tex>w</fr:tex>' are elementarily equivalent)
-        </fr:li>
-        <fr:li>
-            (<fr:em>forth</fr:em>) <fr:tex>w Z w' ∧ w R v ⟹ ∃ v' ∈ M'. w' R' v' ∧ v Z v'</fr:tex></fr:li>
-        <fr:li>
-            (<fr:em>backwards</fr:em>) <fr:tex>w Z w' ∧ w' R' v' ⟹ ∃ v ∈ M. w R v ∧ v Z v'</fr:tex></fr:li></fr:ul></fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    That is, (classical) FoL is often the setting in which axiomatic
+    mathematical theories are formalized. Classic (no pun intended) examples
+    are <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{group} = \{e, ⋅, -⁻¹\}, Th(\textsf {groups}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{DLO} = \{≤\}, Th(\textsf {Dense Linear Orders}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{sets} = \{∈\}, ZFC)]]></fr:tex>.
+    (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
+</fr:p><fr:p>
+    "definite property"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:taxon>Moral corollary</fr:taxon><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    "Modal formulas cannot distinguish between bisimilar worlds or between bisimilar Kripke frames"
-
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>.
-    A <fr:em>simulation</fr:em> is where we drop the <fr:em>backwards</fr:em> condition.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
+        properties of logics, type theories, and programming languages:
+        canonicity, decidability, conservativity, computational adequacy, and
+        more. Logical relations combine pure syntax with non-syntactic objects
+        that are parameretized in syntax in a somewhat complex way; the
+        sophistication of possible parameterizations makes logical relations a
+        tool that is primarily accessible to specialists. In the spirit of
+        Halmos’ Naïve Set Theory, we advance a new viewpoint on logical
+        relations based on synthetic Tait computability, an internal language
+        for categories of logical relations. In <fr:strong>synthetic Tait
+        computability</fr:strong>, logical relations are manipulated as if they were sets,
+        making the essence of many complex logical relations arguments
+        accessible to non-specialists.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:taxon>Corollary</fr:taxon><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
+        idea of what's going on starting at p. 9.
+    </fr:p><fr:p>
+        From what I can tell, the idea is to work in the internal language of some
+        topos-- really the internal "agda-like dependent type theory"-- to
+        construct the logical relations category, instead of doing something like
+        (analytic) gluing.
+        How this makes anything easier is beyond me at the moment.
+        I can't tell if the <fr:em>object-space/meta-space</fr:em> discussion is clear
+        and I just don't have some context, or it's actually presented almost
+        mystically.
+        The mention of <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    Clearly the truth of <fr:tex>∧,∨,◇</fr:tex>-formulas are preserved under simulations.
-    Actually it's shown that the class of formulas preserved under simulations are exactly these formulas.
-</fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
-</fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+href="carette-kiselyov-shan-2007.xml"
+addr="carette-kiselyov-shan-2007"
+title="Finally Tagless, Partially Evaluated">finally tagless encodings</fr:link>
+        and <fr:link
+type="external"
+href="hoffman-1999">a justification for hoas</fr:link> doesn't seem super
+        on-the-point, but if it's mentioned it must be for good reason…
+    </fr:p><fr:p>
+        Again, quotes are air-quotes, not actual quotes.
+        <fr:link
+type="external"
+href="sterling-2022.pdf">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>801</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-    
-    <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>271</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Short About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    I am a master's student in Computer Science and Engineering at the
-    University of Michigan.
-    I'm interested in logic and related areas of computer science, particularly
-    programming languages.
-</fr:p><fr:p>
-    This is my <fr:link
-href="https://www.jonmsterling.com/jms-005P.xml"
-type="external">forest</fr:link>.
-</fr:p><fr:ul><fr:li>
-                email: <fr:embedded-tex
-hash="981fdd9f85dd838fc676b7448e3e9273"><fr:embedded-tex-preamble></fr:embedded-tex-preamble><fr:embedded-tex-body>hejohns@umich.edu</fr:embedded-tex-body></fr:embedded-tex></fr:li>
-    <fr:li>
-        photo:
-        <fr:p><html:img
-xmlns:html="http://www.w3.org/1999/xhtml"
-src="hejohns.jpeg"
-width="150"
-height="150"></html:img></fr:p></fr:li>
-    <fr:li>
-        cv: <fr:link
-href="cv.pdf"
-type="external">cv.pdf</fr:link> (TODO)
-    </fr:li></fr:ul></fr:mainmatter></fr:tree>
-<fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
+        resource consciousness that makes sense in classical predicate logic.
+        We characterize the propositional formulas all of whose first-order
+        instances are simply Herbrand valid. The characterization turns out to
+        coincide with a known characterization of game semantical validity for
+        multiplicative formulas.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    This is currently a "would like to read, but no urgency" paper, because
+    Blass + Herbrand's theorem.
+    <fr:link
+type="external"
+href="blass-2003.pdf">blass-2003.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
 show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>273</fr:anchor><fr:title>Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    What is the actual <fr:em>logic</fr:em> when doing a logical relation?
+    Do you only actually need functors? Zeilberger's <fr:link
+type="external"
+href="millies-zeilberger-2015.pdf">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
+    Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
+</fr:p><fr:p>
+    Talked to Max about this on 5-9.
+    They really just give to different notions of "logic over syntax"--
+    a hoare logic, and a implicational (eg FO) logic.
+    I suppose one'd want to study a bit the relationship between the two.
+    <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        My screen name is <fr:code>hejohns</fr:code>, eg <fr:link
-href="https://github.com/hejohns"
-type="external">https://github.com/hejohns</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="millies-zeilberger-2015.xml"
+addr="millies-zeilberger-2015"
+title="Functors are Type Refinement Systems">paper</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
+</fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/jacobs-1999.xml b/_site/forest/jacobs-1999.xml
index 093afba..131c1cd 100644
--- a/_site/forest/jacobs-1999.xml
+++ b/_site/forest/jacobs-1999.xml
@@ -1,13 +1,14 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>723</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>916</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/johnstone-2002.xml b/_site/forest/johnstone-2002.xml
index 8612328..7ec9dc0 100644
--- a/_site/forest/johnstone-2002.xml
+++ b/_site/forest/johnstone-2002.xml
@@ -1,245 +1,381 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>657</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>484</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>658</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -247,7 +383,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -255,131 +392,589 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
+    </fr:p><fr:p>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>659</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+type="local"
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
+        classifier and a natural numbers object."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+type="local"
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
+type="local"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course, if we ignore natural numbers objects for now.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/lambek-scott-1986.xml b/_site/forest/lambek-scott-1986.xml
index 8e5fbcc..12d13b1 100644
--- a/_site/forest/lambek-scott-1986.xml
+++ b/_site/forest/lambek-scott-1986.xml
@@ -1,10 +1,980 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>450</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>554</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    There is some (a posteriori insignificant) variation in the presentation of
+    the syntax, thus following through with some input from <fr:link
+type="local"
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
+    juxtapose multiple definitions-- <fr:link
+type="local"
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
+    topoi</fr:link>--, but this time we slightly modify the source for
+    uniformity as needed.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
+type="local"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
+        <fr:ul><fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
+            <fr:li>
+                a class of <fr:em>terms</fr:em> of each type, including countably many
+                variables of each type
+            </fr:li>
+            <fr:li>
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
+            </fr:li></fr:ul>"
+    </fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The class of types is closed under the inductive clauses:
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The class of terms is freely generated from certain basic terms by
+            certain operations including the following… and is closed under certain operations.
+            <fr:ul><fr:li>
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
+                </fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
+        </fr:p><fr:p>
+            "One may also define a number of widely used term forming
+            operations: …
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (We write "language" whenever Johnstone writes "signature", and use the
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
+        I don't know if Johnstone prefers the term "signature" because
+        "language" is overloaded and often refers not just to the, well,
+        language, but to the larger logic sans deductive system, but that would
+        be a good reason.)
+    </fr:p><fr:p>
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        of <fr:em>type constructors</fr:em> chosen from the following list, which
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
+    </fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+                (i) <fr:em>Basic types</fr:em>: Each sort is a type
+            </fr:li>
+            <fr:li>
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
+                type.
+            </fr:li>
+            <fr:li>
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
+                a type.
+            </fr:li>
+            <fr:li>
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
+            </fr:li>
+            <fr:li>
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
+            </fr:li></fr:ul><fr:p>
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
+        </fr:p><fr:p>
+            "If our language has the type constructors (ii) and (iv), we shall
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
+        </fr:p><fr:p>
+            "From now on, all our languages will contain the product type
+            constructor; thus we have restricted all our primitive function and
+            relation symbols to be unary. (If our signature also contaisn
+            function types, then we can further restrict our function symbols
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
+            primitive relation symbols by constants as well)"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+        "The term constructor for power types in fact builds terms form
+        formulae, so over a language which contains power types we have to
+        define terms and formulae by a simultaneous recursion".
+    </fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            (We omit the bookkeeping of free variables that Johnstone inlines with
+            the following definitions.)
+            "<fr:ul><fr:li>
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (iii)
+                    <fr:ul><fr:li>
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
+                        </fr:li>
+                        <fr:li>
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
+                        </fr:li></fr:ul></fr:li>
+                <fr:li>
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
+                    [This is formally different from (ii).]
+                </fr:li>
+                <fr:li>
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (viii)
+                    <fr:ul><fr:li>
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
+                        </fr:li>
+                        <fr:li>
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
+                        </fr:li></fr:ul></fr:li>
+                <fr:li>
+                    (ix) The remaining term constructor associated with list
+                    types, the <fr:em>iterator</fr:em>, is more complicated [which we
+                    omit for irrelevance].
+                </fr:li></fr:ul></fr:p><fr:p>
+            Of course, clauses (v) and (vi) of the definition above apply only
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
+            types, and clauses (viii) and (ix) only if we have list types.
+            Note that, except for power types, the clauses come in pairs [of
+            constructors and destructors]. The reason why we have no
+            'destructor' for power types is that, just as the constructor for
+            power types builds terms out of formulae rather than terms, the
+            corresponding destructor builds formulae out of terms of a power
+            type-- it thus appears as clause (iii) of the next definition."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            (We again omit the bookkeeping of free variables.)
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
+            <fr:ul><fr:li>
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
+                </fr:li>
+                <fr:li>
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
+                </fr:li>
+                <fr:li>
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
+                </fr:li></fr:ul></fr:p><fr:p>
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
+        interpretation), instead of describing the logic abstractly and taking
+        the internal language to be a diagram (in Tarskian-model theory speak)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+    </fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            The types have the structure of the objects of an elementary topos,
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The [language] is sorted…. We inductively define terms and their sorts:
+            <fr:ul><fr:li>
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
+                </fr:li>
+                <fr:li>
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
+                </fr:li></fr:ul>"
+            <fr:ul><fr:li>
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
+            these terms in any elementary topos, developed in 13.3 and 13.4],
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
+        </fr:p><fr:p>
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Ignoring the natural numbers type and the idiosyncratic list type of
+        <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+        elementary topops for reference) of <fr:link
+type="local"
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
+        predicate logic" aka "topos logic" aka "higher-order intuitionistic
+        logic"</fr:link>, although certain inter-definable choices can be
+        made.
+    </fr:p><fr:p>
+        There's no clear notion of "taking everything", but a reasonable
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
+    </fr:p><fr:p>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>556</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+type="local"
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
+        classifier and a natural numbers object."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+type="local"
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
+type="local"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course, if we ignore natural numbers objects for now.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/lawvere-1970.xml b/_site/forest/lawvere-1970.xml
index dd7c97a..90127fb 100644
--- a/_site/forest/lawvere-1970.xml
+++ b/_site/forest/lawvere-1970.xml
@@ -1,20 +1,20 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>497</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>647</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -44,8 +44,11 @@ root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:ta
         and transformation of the thing. Doing this for "set theory" requires
         taking account of the experience that the main pairs of opposing
         tendencies in mathematics take the form of adjoint functors, and frees
-        us of the mathematically irrelevant traces (<fr:tex>∈</fr:tex>) left behind by the
-        process of accumulating (<fr:tex>∪</fr:tex>) the power set (<fr:tex>\mathcal {P}</fr:tex>) at each stage of a
+        us of the mathematically irrelevant traces (<fr:tex
+display="inline"><![CDATA[∈]]></fr:tex>) left behind by the
+        process of accumulating (<fr:tex
+display="inline"><![CDATA[∪]]></fr:tex>) the power set (<fr:tex
+display="inline"><![CDATA[\mathcal {P}]]></fr:tex>) at each stage of a
         metaphysical "construction". Further, experience with sheaves,
         permutation representations, algebraic spaces, etc., shows that a "set
         theory" for geometry should apply not only to abstract sets divorced
@@ -53,32 +56,43 @@ root="false"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:taxon>Abstract</fr:ta
         sets which do in fact develop along such parameters. For such sets,
         usually logic is "intuitionistic" (in its formal properties)
         usually the axiom of choice is false, and usually a set is not
-        determined by its points defined over <fr:tex>1</fr:tex> only.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+        determined by its points defined over <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> only.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     I have not read this, and have no idea what "contradictions" he's referring to.
     <fr:link
-href="lawvere-1970.pdf"
-type="external">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="lawvere-1970.pdf">lawvere-1970.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>495</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>648</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -92,9 +106,10 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/logicmatters.xml b/_site/forest/logicmatters.xml
index bb39b61..c6f4d6f 100644
--- a/_site/forest/logicmatters.xml
+++ b/_site/forest/logicmatters.xml
@@ -1,32 +1,40 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>logicmatters</fr:addr><fr:route>logicmatters.xml</fr:route><fr:title>Peter Smith's Book Reviews</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>525</fr:anchor><fr:addr>logicmatters</fr:addr><fr:route>logicmatters.xml</fr:route><fr:title>Peter Smith's Book Reviews</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of my all-time favorite pages on the web (and that's not an uncommonly
     held opinion), a tremendous resource and service for the community. I'm not
     able to evaluate too many of Prof. Smith's opinions, but I don't disagree
     with any of what he's written about the books I have seen.
 </fr:p><fr:p>
     At least I just reread the review of <fr:link
-href="shoenfield-1967.xml"
 type="local"
-addr="shoenfield-1967">Mathematical Logic</fr:link>, and "So this
+href="shoenfield-1967.xml"
+addr="shoenfield-1967"
+title="Mathematical Logic">Mathematical Logic</fr:link>, and "So this
     neither has the cleanness of a Hilbert system nor the naturalness of a
     natural deduction system. As far as I noticed, nothing is said to motivate
     this seemingly horrible choice as against others." always makes me laugh.
 </fr:p><fr:p><fr:link
-href="https://www.logicmatters.net/tyl/booknotes/"
-type="external">https://www.logicmatters.net/tyl/booknotes/</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://www.logicmatters.net/tyl/booknotes/">https://www.logicmatters.net/tyl/booknotes/</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>526</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/maclane-moerdijk-1994.xml b/_site/forest/maclane-moerdijk-1994.xml
index 0ac2f1b..3573186 100644
--- a/_site/forest/maclane-moerdijk-1994.xml
+++ b/_site/forest/maclane-moerdijk-1994.xml
@@ -1,32 +1,436 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>483</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>652</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>653</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title><fr:link
+type="local"
+href="unstable-199.xml"
+addr="#199"
+title="What is an elementary topos? › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory"><fr:link
+type="local"
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">What is an elementary topos?</fr:link> › Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>655</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+type="local"
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
+        classifier and a natural numbers object."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+type="local"
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
+type="local"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course, if we ignore natural numbers objects for now.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/maddy-2019.xml b/_site/forest/maddy-2019.xml
index 2261055..64b8179 100644
--- a/_site/forest/maddy-2019.xml
+++ b/_site/forest/maddy-2019.xml
@@ -1,20 +1,20 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>572</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>765</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -30,7 +30,7 @@ root="false"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:taxon>Abstract</fr:ta
         done. What matters is these jobs we want our theories to do and how well
         they do them. Whether any or all of them, jobs or theories, deserves to be
         called ‘foundational’ is really beside the point.
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     A nice article that can be briefly summarized as
     <fr:ul><fr:li>
             There is no one clear set of desiderata for a <fr:em>foundational</fr:em>
@@ -64,5 +64,5 @@ root="false"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:taxon>Abstract</fr:ta
 </fr:p><fr:p>
     I've used quotes here as air quotes, not actual quotes of Maddy.
     <fr:link
-href="maddy-2019.pdf"
-type="external">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+type="external"
+href="maddy-2019.pdf">maddy-2019.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/makkai-reyes-1977.xml b/_site/forest/makkai-reyes-1977.xml
index 5ee64a4..954799d 100644
--- a/_site/forest/makkai-reyes-1977.xml
+++ b/_site/forest/makkai-reyes-1977.xml
@@ -1,13 +1,398 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>480</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>644</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>645</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+type="local"
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
+        classifier and a natural numbers object."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+type="local"
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
+type="local"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course, if we ignore natural numbers objects for now.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/marquis-reyes-2012.xml b/_site/forest/marquis-reyes-2012.xml
index cced133..435d902 100644
--- a/_site/forest/marquis-reyes-2012.xml
+++ b/_site/forest/marquis-reyes-2012.xml
@@ -1,48 +1,52 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>507</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>664</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
-href="ronnie-chen.xml"
 type="local"
-addr="ronnie-chen">Ronnie Chen</fr:link>,
+href="ronnie-chen.xml"
+addr="ronnie-chen"
+title="Ronnie Chen">Ronnie Chen</fr:link>,
     where the viewpoint of "generalized algebraic semantics" is developed.
 </fr:p><fr:p><fr:link
-href="marquis-reyes-2012.pdf"
-type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+type="external"
+href="marquis-reyes-2012.pdf">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>506</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>ronnie-chen</fr:addr><fr:route>ronnie-chen.xml</fr:route><fr:title>Ronnie Chen</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>504</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>665</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -56,9 +60,24 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>667</fr:anchor><fr:addr>ronnie-chen</fr:addr><fr:route>ronnie-chen.xml</fr:route><fr:title>Ronnie Chen</fr:title><fr:taxon>Person</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/mclarty-1990a.xml b/_site/forest/mclarty-1990a.xml
index 02f760a..0a5103c 100644
--- a/_site/forest/mclarty-1990a.xml
+++ b/_site/forest/mclarty-1990a.xml
@@ -1,12 +1,12 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>494</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="mclarty-1990a.pdf"
-type="external">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>640</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/mclarty-1990b.xml b/_site/forest/mclarty-1990b.xml
index b2cbde0..8385958 100644
--- a/_site/forest/mclarty-1990b.xml
+++ b/_site/forest/mclarty-1990b.xml
@@ -1,20 +1,20 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>503</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>661</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -25,62 +25,78 @@ root="false"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:taxon>Abstract</fr:ta
         remarks on a categorical view of the history of set theory, including a
         false history plausible from that point of view that would make it helpful
         to introduce toposes as a generalization from set theory.
-    </fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="mclarty-1990b.pdf"
-type="external">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="mclarty-1990b.pdf">mclarty-1990b.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>501</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>662</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -96,107 +112,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -211,121 +250,166 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/mclarty-1992.xml b/_site/forest/mclarty-1992.xml
index b65735c..a6ce2f5 100644
--- a/_site/forest/mclarty-1992.xml
+++ b/_site/forest/mclarty-1992.xml
@@ -1,106 +1,47 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>544</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>701</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>537</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000L</fr:addr><fr:route>hejohns-000L.xml</fr:route><fr:title>Generalized element</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A <fr:em>generalized element</fr:em> (alt. local element) of an object <fr:tex>A ∈      \mathcal {C}</fr:tex> is <fr:em>just</fr:em> a morphism <fr:tex>e : B → A</fr:tex>, for some <fr:tex>B</fr:tex>. Ie, a
-    generalized element of <fr:tex>A</fr:tex> is <fr:em>just</fr:em> an object of the slice over <fr:tex>A</fr:tex>.
-</fr:p><fr:p>
-    There's supposed to be some Kripke-style interpretation of these as
-    elements defined at points in time, but TODO.
-</fr:p><fr:p>
-    From <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
-href="hejohns-000K.xml"
-type="local"
-addr="hejohns-000K">global element</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>538</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000K</fr:addr><fr:route>hejohns-000K.xml</fr:route><fr:title>Global element</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A <fr:em>global element</fr:em> of an object <fr:tex>A ∈  \mathcal {C}</fr:tex> is <fr:em>just</fr:em> a
-    morphism <fr:tex>e : 1 → A</fr:tex>.
-</fr:p><fr:p>
-    From <fr:link
-href="mclarty-1992.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>702</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
-href="hejohns-000L.xml"
-type="local"
-addr="hejohns-000L">generalized element</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>539</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
-    <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
-    is a pullback square.
-</fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
-type="local"
-addr="hejohns-000K">global
-    element</fr:link> is trivially monic. Recall that "monos are stable
-    under pullback".
-</fr:p><fr:p>
-    This is from <fr:link
-href="mclarty-1992.xml"
-type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>540</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -121,27 +62,614 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>704</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    There is some (a posteriori insignificant) variation in the presentation of
+    the syntax, thus following through with some input from <fr:link
+type="local"
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
+    juxtapose multiple definitions-- <fr:link
+type="local"
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
+    topoi</fr:link>--, but this time we slightly modify the source for
+    uniformity as needed.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>542</fr:anchor><fr:taxon>Exposition</fr:taxon><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
+        <fr:ul><fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
+            <fr:li>
+                a class of <fr:em>terms</fr:em> of each type, including countably many
+                variables of each type
+            </fr:li>
+            <fr:li>
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
+            </fr:li></fr:ul>"
+    </fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The class of types is closed under the inductive clauses:
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The class of terms is freely generated from certain basic terms by
+            certain operations including the following… and is closed under certain operations.
+            <fr:ul><fr:li>
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
+                </fr:li>
+                <fr:li>
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
+                </fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
+        </fr:p><fr:p>
+            "One may also define a number of widely used term forming
+            operations: …
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
+                </fr:li>
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (We write "language" whenever Johnstone writes "signature", and use the
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
+        I don't know if Johnstone prefers the term "signature" because
+        "language" is overloaded and often refers not just to the, well,
+        language, but to the larger logic sans deductive system, but that would
+        be a good reason.)
+    </fr:p><fr:p>
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        of <fr:em>type constructors</fr:em> chosen from the following list, which
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
+    </fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+                (i) <fr:em>Basic types</fr:em>: Each sort is a type
+            </fr:li>
+            <fr:li>
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
+                type.
+            </fr:li>
+            <fr:li>
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
+                a type.
+            </fr:li>
+            <fr:li>
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
+            </fr:li>
+            <fr:li>
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
+            </fr:li></fr:ul><fr:p>
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
+        </fr:p><fr:p>
+            "If our language has the type constructors (ii) and (iv), we shall
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
+        </fr:p><fr:p>
+            "From now on, all our languages will contain the product type
+            constructor; thus we have restricted all our primitive function and
+            relation symbols to be unary. (If our signature also contaisn
+            function types, then we can further restrict our function symbols
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
+            primitive relation symbols by constants as well)"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+        "The term constructor for power types in fact builds terms form
+        formulae, so over a language which contains power types we have to
+        define terms and formulae by a simultaneous recursion".
+    </fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            (We omit the bookkeeping of free variables that Johnstone inlines with
+            the following definitions.)
+            "<fr:ul><fr:li>
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (iii)
+                    <fr:ul><fr:li>
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
+                        </fr:li>
+                        <fr:li>
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
+                        </fr:li></fr:ul></fr:li>
+                <fr:li>
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
+                    [This is formally different from (ii).]
+                </fr:li>
+                <fr:li>
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (viii)
+                    <fr:ul><fr:li>
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
+                        </fr:li>
+                        <fr:li>
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
+                        </fr:li></fr:ul></fr:li>
+                <fr:li>
+                    (ix) The remaining term constructor associated with list
+                    types, the <fr:em>iterator</fr:em>, is more complicated [which we
+                    omit for irrelevance].
+                </fr:li></fr:ul></fr:p><fr:p>
+            Of course, clauses (v) and (vi) of the definition above apply only
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
+            types, and clauses (viii) and (ix) only if we have list types.
+            Note that, except for power types, the clauses come in pairs [of
+            constructors and destructors]. The reason why we have no
+            'destructor' for power types is that, just as the constructor for
+            power types builds terms out of formulae rather than terms, the
+            corresponding destructor builds formulae out of terms of a power
+            type-- it thus appears as clause (iii) of the next definition."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            (We again omit the bookkeeping of free variables.)
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
+            <fr:ul><fr:li>
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
+                </fr:li>
+                <fr:li>
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
+                </fr:li>
+                <fr:li>
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
+                </fr:li></fr:ul></fr:p><fr:p>
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
+        interpretation), instead of describing the logic abstractly and taking
+        the internal language to be a diagram (in Tarskian-model theory speak)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+    </fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            The types have the structure of the objects of an elementary topos,
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "The [language] is sorted…. We inductively define terms and their sorts:
+            <fr:ul><fr:li>
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
+                </fr:li>
+                <fr:li>
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
+                </fr:li>
+                <fr:li>
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
+                </fr:li></fr:ul>"
+            <fr:ul><fr:li>
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
+            these terms in any elementary topos, developed in 13.3 and 13.4],
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
+        </fr:p><fr:p>
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Ignoring the natural numbers type and the idiosyncratic list type of
+        <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+        elementary topops for reference) of <fr:link
+type="local"
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
+        predicate logic" aka "topos logic" aka "higher-order intuitionistic
+        logic"</fr:link>, although certain inter-definable choices can be
+        made.
+    </fr:p><fr:p>
+        There's no clear notion of "taking everything", but a reasonable
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
+    </fr:p><fr:p>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>705</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>225</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title>Elementary questions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Some basic questions, whose "elevator pitch" answers ought to roll off
         the tongue:
         <fr:ul><fr:li>
@@ -162,9 +690,10 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
                 Is it the same thing as "topos logic"?
             </fr:li></fr:ul>
         Following <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>, we sketch <fr:em>elementary</fr:em> answers to these.
     </fr:p><fr:p>
         Some more difficult questions we do not attempt to answer at the moment:
         <fr:ul><fr:li>
@@ -191,37 +720,41 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>, we sket
         complete the sentence".
         A truly FAQ-style answer compendium will have to come later-- after I've
         actually learned the basics.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>309</fr:anchor><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>227</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title>Elementary answers</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000C</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>229</fr:anchor><fr:addr>hejohns-000H</fr:addr><fr:route>hejohns-000H.xml</fr:route><fr:title>Why is it called an "elementary" topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="blass-1988.xml"
-type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "The word 'elementary' in connection with the Lawvere-Tierney definition refers to the fact that their axioms are expressed in the first-order language of categories; in fact the axioms are essentially algebraic." (p. 2).
     (Recall that categories can be formalized as a two or dependently-sorted
     first order theory.)
 </fr:p><fr:p>
     This is corroborated by <fr:link
-href="marquis-reyes-2012.xml"
 type="local"
-addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
+href="marquis-reyes-2012.xml"
+addr="marquis-reyes-2012"
+title="The history of categorical logic: 1963-1977">The history of categorical logic: 1963-1977</fr:link>, writing that "In the years
     that followed, Lawvere tried to extend his analysis and sketched a
     categorical version of first-order theories under the name of elementary
     theories. Then, in 1969, in collaboration with Myles Tierney, Lawvere
@@ -235,65 +768,75 @@ addr="marquis-reyes-2012">The history of categorical logic: 1963-1977</fr:link>,
     differential geometry, constructive mathematics were made." (p. 2).
 </fr:p><fr:p>
     If <fr:link
-href="lawvere-1970.xml"
 type="local"
-addr="lawvere-1970">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
+href="lawvere-1970.xml"
+addr="lawvere-1970"
+title="Quantifiers and Sheaves">Quantifiers and Sheaves</fr:link> is indeed the paper that introduces elementary toposes,
     it's curious to note that the word "elementary" never appears.
     It must've been coined sometime later, but I don't know the details.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>231</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -309,107 +852,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -424,148 +990,196 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:addr>hejohns-000F</fr:addr><fr:route>hejohns-000F.xml</fr:route><fr:title>"Higher-order intuitionistic logic" v. "topos logic"</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Many authors object to the name "higher-order intuitionistic logic".
 </fr:p><fr:p><fr:link
-href="blass-1988.xml"
 type="local"
-addr="blass-1988">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
+href="blass-1988.xml"
+addr="blass-1988"
+title="Topoi and Computation">Topoi and Computation</fr:link> writes that "Strictly speaking, the phrase 'higher-order
     intuitionistic' makes little sense, since it is not clear that
     constructions like the power set are intuitionistically acceptable. What is
     meant by the phrase is a type theory built on a formally intuitionistic
     underlying logic." (p. 2).
 </fr:p><fr:p>
     And <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> similarly writes "In Part III the first order topos
     axioms are used to define the higher order internal language. The logic is
     called 'topos logic' here, although 'intuitionistic logic' is often
     referred to in the literature. I believe that 'intuitionism' is usually,
@@ -586,248 +1200,377 @@ addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> similarl
     Although I like calling it "finite-type intuitionistic predicate logic" to
     keep in touch with some older terminology you still see around the block.
 
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:addr>hejohns-000J</fr:addr><fr:route>hejohns-000J.xml</fr:route><fr:title>Higher-order intuitionistic topos logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Syntax</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
-href="ericgio.xml"
 type="local"
-addr="ericgio">Eric Giovannini</fr:link>, we
+href="ericgio.xml"
+addr="ericgio"
+title="Eric Giovannini">Eric Giovannini</fr:link>, we
     juxtapose multiple definitions-- <fr:link
-href="hejohns-000G.xml"
 type="local"
-addr="hejohns-000G">as done for elementary
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">as done for elementary
     topoi</fr:link>--, but this time we slightly modify the source for
     uniformity as needed.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="lambek-scott-1986.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.128</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>type theory</fr:em> is a formal theory given by the following data:
         <fr:ul><fr:li>
-                a class of <fr:em>types</fr:em>, including a special type <fr:tex>Ω</fr:tex></fr:li>
+                a class of <fr:em>types</fr:em>, including a special type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex></fr:li>
             <fr:li>
                 a class of <fr:em>terms</fr:em> of each type, including countably many
                 variables of each type
             </fr:li>
             <fr:li>
-                for each finite set <fr:tex>X</fr:tex> of variables a binary relation <fr:tex>⊢_X</fr:tex>
-                of <fr:em>entailment</fr:em> between terms of type <fr:tex>Ω</fr:tex> all free
-                variables of which are elements of <fr:tex>X</fr:tex>.
+                for each finite set <fr:tex
+display="inline"><![CDATA[X]]></fr:tex> of variables a binary relation <fr:tex
+display="inline"><![CDATA[⊢_X]]></fr:tex>
+                of <fr:em>entailment</fr:em> between terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> all free
+                variables of which are elements of <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>.
             </fr:li></fr:ul>"
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>246</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of types is closed under the inductive clauses:
-            <fr:ul><fr:li><fr:tex>1</fr:tex>, <fr:tex>N</fr:tex> and <fr:tex>Ω</fr:tex> are types ('basic' types);
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[1]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are types ('basic' types);
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, so are <fr:tex>A×B</fr:tex> and <fr:tex>\mathcal {P} A</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, so are <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The class of terms is freely generated from certain basic terms by
             certain operations including the following… and is closed under certain operations.
             <fr:ul><fr:li>
-                    [for each type <fr:tex>A</fr:tex>, there are countably many variables
-                    <fr:tex>x_1^A, x_2^A, …</fr:tex>;]
+                    [for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there are countably many variables
+                    <fr:tex
+display="inline"><![CDATA[x_1^A, x_2^A, …]]></fr:tex>;]
                 </fr:li>
-                <fr:li><fr:tex>*</fr:tex> is a term of type <fr:tex>1</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[*]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>b</fr:tex> is a term of type
-                    <fr:tex>B</fr:tex>, then <fr:tex>⟨a,b⟩</fr:tex> is a term of type <fr:tex>A×B</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[b]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨a,b⟩]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>a</fr:tex> is a term of type <fr:tex>A</fr:tex> and <fr:tex>α</fr:tex> is a term of type
-                    <fr:tex>\mathcal {P} a</fr:tex>, then <fr:tex>a ∈ α</fr:tex> is a term of type <fr:tex>Ω</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[α]]></fr:tex> is a term of type
+                    <fr:tex
+display="inline"><![CDATA[\mathcal {P}a]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[a ∈ α]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>\{ x ∈ A | ϕ(x) \}</fr:tex>
-                    is a term of type <fr:tex>\mathcal {P} A</fr:tex> not containing a free occurrence of
-                    <fr:tex>x</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[\{x ∈ A | ϕ(x)\}]]></fr:tex>
+                    is a term of type <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> not containing a free occurrence of
+                    <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>0</fr:tex> is a term of type <fr:tex>N</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[0]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>n</fr:tex> is a term of type <fr:tex>N</fr:tex> so is <fr:tex>Sn</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[n]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> so is <fr:tex
+display="inline"><![CDATA[Sn]]></fr:tex>;
                 </fr:li>
-                <fr:li><fr:tex>⊤</fr:tex> and <fr:tex>⊥</fr:tex> are terms of type <fr:tex>Ω</fr:tex>;
+                <fr:li><fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[⊥]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>p</fr:tex> and <fr:tex>q</fr:tex> are terms of type <fr:tex>Ω</fr:tex>, so are <fr:tex>p∧q</fr:tex>,
-                    <fr:tex>p∨q</fr:tex>, <fr:tex>p⟹q</fr:tex>;
+                    if <fr:tex
+display="inline"><![CDATA[p]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[q]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, so are <fr:tex
+display="inline"><![CDATA[p∧q]]></fr:tex>,
+                    <fr:tex
+display="inline"><![CDATA[p∨q]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[p⟹q]]></fr:tex>;
                 </fr:li>
                 <fr:li>
-                    if <fr:tex>ϕ(x)</fr:tex> is a term of type <fr:tex>Ω</fr:tex> (possibly containing the
-                    free variable <fr:tex>x</fr:tex> of type <fr:tex>A</fr:tex>), then <fr:tex>∀_{x∈A}ϕ(x)</fr:tex> and
-                    <fr:tex>∃_{x∈A}ϕ(x)</fr:tex> are terms of type <fr:tex>Ω</fr:tex> not containing free
-                    occurrences of <fr:tex>x</fr:tex>.
+                    if <fr:tex
+display="inline"><![CDATA[ϕ(x)]]></fr:tex> is a term of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (possibly containing the
+                    free variable <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), then <fr:tex
+display="inline"><![CDATA[∀_{x∈A}ϕ(x)]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[∃_{x∈A}ϕ(x)]]></fr:tex> are terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> not containing free
+                    occurrences of <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>.
                 </fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>248</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "Terms of type <fr:tex>Ω</fr:tex> are also called <fr:em>formulas</fr:em>."
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>259</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#179</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "Terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> are also called <fr:em>formulas</fr:em>."
         </fr:p><fr:p>
             "One may also define a number of widely used term forming
             operations: …
-            <fr:ul><fr:li><fr:tex>p ⟺ q ≡ (p⟹q)∧(q⟹q)</fr:tex>,
+            <fr:ul><fr:li><fr:tex
+display="inline"><![CDATA[p ⟺ q ≡ (p⟹q)∧(q⟹q)]]></fr:tex>,
                 </fr:li>
-                <fr:li><fr:tex>a = a' ≡ ∀_{u∈ \mathcal {P} A }(a∈u ⟺ a'∈u)</fr:tex></fr:li></fr:ul>"
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                <fr:li><fr:tex
+display="inline"><![CDATA[a = a' ≡ ∀_{u∈\mathcal {P}A}(a∈u ⟺ a'∈u)]]></fr:tex></fr:li></fr:ul>"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>253</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="johnstone-2002.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>260</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> </fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         (We write "language" whenever Johnstone writes "signature", and use the
-        corresponding symbol <fr:tex>\mathcal {L}</fr:tex> instead of <fr:tex>Σ</fr:tex>.
+        corresponding symbol <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> instead of <fr:tex
+display="inline"><![CDATA[Σ]]></fr:tex>.
         I don't know if Johnstone prefers the term "signature" because
         "language" is overloaded and often refers not just to the, well,
         language, but to the larger logic sans deductive system, but that would
         be a good reason.)
     </fr:p><fr:p>
-        "A <fr:em>language</fr:em> <fr:tex>\mathcal {L}</fr:tex> for higher-order logic is defined by specifying a
-        set <fr:tex>\mathcal {L} _{ \mathsf {base}}</fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex>\mathcal {L} _{ \mathsf {fun}}</fr:tex> of <fr:em>function symbols</fr:em> and
-        a set <fr:tex>\mathcal {L} _{ \mathsf {rel}}</fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
+        "A <fr:em>language</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> for higher-order logic is defined by specifying a
+        set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {base}}]]></fr:tex> of <fr:em>atomic types</fr:em>, a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {fun}}]]></fr:tex> of <fr:em>function symbols</fr:em> and
+        a set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {rel}}]]></fr:tex> of <fr:em>relation symbols</fr:em> as in 1.1.1, together with a number
         of <fr:em>type constructors</fr:em> chosen from the following list, which
-        recursively define the set <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex> of <fr:tex>types</fr:tex>:
+        recursively define the set <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[types]]></fr:tex>:
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>250</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>261</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
                 (i) <fr:em>Basic types</fr:em>: Each sort is a type
             </fr:li>
             <fr:li>
-                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex>1</fr:tex> (the
-                empty product); and if <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A×B</fr:tex>is a
+                (ii) <fr:em>Product types</fr:em>: There is a distinguished type <fr:tex
+display="inline"><![CDATA[1]]></fr:tex> (the
+                empty product); and if <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>is a
                 type.
             </fr:li>
             <fr:li>
-                (iii) <fr:em>Function types</fr:em>: If <fr:tex>A</fr:tex> and <fr:tex>B</fr:tex> are types, then <fr:tex>A→B</fr:tex> is
+                (iii) <fr:em>Function types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> are types, then <fr:tex
+display="inline"><![CDATA[A→B]]></fr:tex> is
                 a type.
             </fr:li>
             <fr:li>
-                (iv) <fr:em>Power types</fr:em>: If <fr:tex>A</fr:tex> is a type, the <fr:tex>\mathcal {P} A</fr:tex> is a type.
+                (iv) <fr:em>Power types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, the <fr:tex
+display="inline"><![CDATA[\mathcal {P}A]]></fr:tex> is a type.
             </fr:li>
             <fr:li>
-                (v) <fr:em>List types</fr:em>: If <fr:tex>A</fr:tex> is a type, then <fr:tex>LA</fr:tex> is a type.
+                (v) <fr:em>List types</fr:em>: If <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a type, then <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex> is a type.
             </fr:li></fr:ul><fr:p>
-            To each function symbol <fr:tex>f∈ \mathcal {L} _{ \mathsf {fun}}</fr:tex> we assign an ordered pair
-            <fr:tex>(A,B)</fr:tex> of elements of <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>; as before, we write <fr:tex>f : A→B</fr:tex> to
-            indicate that <fr:tex>f</fr:tex> has the pair of types <fr:tex>(A,B)</fr:tex>. Similarly, each
-            relation symbol <fr:tex>R∈ \mathcal {L} _{ \mathsf {rel}}</fr:tex> has a type in <fr:tex>\mathcal {L} _{ \mathsf {type}}</fr:tex>, and we write <fr:tex>R ↣             A</fr:tex> to indicate that <fr:tex>R</fr:tex> has type <fr:tex>A</fr:tex>."
+            To each function symbol <fr:tex
+display="inline"><![CDATA[f∈\mathcal {L}_{\mathsf {fun}}]]></fr:tex> we assign an ordered pair
+            <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex> of elements of <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>; as before, we write <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> to
+            indicate that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> has the pair of types <fr:tex
+display="inline"><![CDATA[(A,B)]]></fr:tex>. Similarly, each
+            relation symbol <fr:tex
+display="inline"><![CDATA[R∈\mathcal {L}_{\mathsf {rel}}]]></fr:tex> has a type in <fr:tex
+display="inline"><![CDATA[\mathcal {L}_{\mathsf {type}}]]></fr:tex>, and we write <fr:tex
+display="inline"><![CDATA[R ↣             A]]></fr:tex> to indicate that <fr:tex
+display="inline"><![CDATA[R]]></fr:tex> has type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>."
         </fr:p><fr:p>
             "If our language has the type constructors (ii) and (iv), we shall
-            often write <fr:tex>Ω</fr:tex> for the particular type <fr:tex>\mathcal {P} 1</fr:tex>. Similarly if it has
-            constructors (ii) and (v), we write <fr:tex>N</fr:tex> (the type of natural
-            numbers) for <fr:tex>L1</fr:tex>."
+            often write <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> for the particular type <fr:tex
+display="inline"><![CDATA[\mathcal {P}1]]></fr:tex>. Similarly if it has
+            constructors (ii) and (v), we write <fr:tex
+display="inline"><![CDATA[N]]></fr:tex> (the type of natural
+            numbers) for <fr:tex
+display="inline"><![CDATA[L1]]></fr:tex>."
         </fr:p><fr:p>
             "From now on, all our languages will contain the product type
             constructor; thus we have restricted all our primitive function and
             relation symbols to be unary. (If our signature also contaisn
             function types, then we can further restrict our function symbols
-            to be constants… And if <fr:tex>\mathcal {L}</fr:tex> has power types, then we can replace all
+            to be constants… And if <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> has power types, then we can replace all
             primitive relation symbols by constants as well)"
-        </fr:p></fr:mainmatter></fr:tree><fr:p>
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
         "The term constructor for power types in fact builds terms form
         formulae, so over a language which contains power types we have to
         define terms and formulae by a simultaneous recursion".
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>262</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We omit the bookkeeping of free variables that Johnstone inlines with
             the following definitions.)
             "<fr:ul><fr:li>
-                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex>x : A</fr:tex> for each type <fr:tex>A</fr:tex>.
+                    (i) We have a stock of <fr:em>variables</fr:em> <fr:tex
+display="inline"><![CDATA[x : A]]></fr:tex> for each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (ii) if <fr:tex>f : A→B</fr:tex> is a function symbol of <fr:tex>\mathcal {L}</fr:tex> and <fr:tex>t : A</fr:tex>,
-                    then <fr:tex>f t : B</fr:tex>.
+                    (ii) if <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex> is a function symbol of <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>,
+                    then <fr:tex
+display="inline"><![CDATA[f t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (iii)
                     <fr:ul><fr:li>
-                            (a) we have a distinguished term <fr:tex>* : 1</fr:tex>.
+                            (a) we have a distinguished term <fr:tex
+display="inline"><![CDATA[* : 1]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            (b) If <fr:tex>s : A</fr:tex> and <fr:tex>t : B</fr:tex>, then <fr:tex>⟨s,t⟩:A×B</fr:tex>.
+                            (b) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[⟨s,t⟩:A×B]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
-                    (iv) If <fr:tex>t : A×B</fr:tex>, then <fr:tex>π₁ t : A</fr:tex> and <fr:tex>π₂ t : B</fr:tex>.
+                    (iv) If <fr:tex
+display="inline"><![CDATA[t : A×B]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[π₁ t : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[π₂ t : B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (v) If <fr:tex>t : B</fr:tex> and <fr:tex>x</fr:tex> is a variable of type <fr:tex>A</fr:tex>, then
-                    <fr:tex>(λx : A. t) : A→B</fr:tex>.
+                    (v) If <fr:tex
+display="inline"><![CDATA[t : B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[(λx : A. t) : A→B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (vi) If <fr:tex>s : A→B</fr:tex> and <fr:tex>t : A</fr:tex>, then <fr:tex>s t : B</fr:tex>.
+                    (vi) If <fr:tex
+display="inline"><![CDATA[s : A→B]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s t : B]]></fr:tex>.
                     [This is formally different from (ii).]
                 </fr:li>
                 <fr:li>
-                    (vii) If <fr:tex>ϕ</fr:tex> is a formula and <fr:tex>x</fr:tex> is a variable of type
-                    <fr:tex>A</fr:tex>, then <fr:tex>\{ x : A | ϕ \}  :  \mathcal {P} A</fr:tex>.
+                    (vii) If <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> is a formula and <fr:tex
+display="inline"><![CDATA[x]]></fr:tex> is a variable of type
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[\{x : A | ϕ\} : \mathcal {P}A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
                     (viii)
                     <fr:ul><fr:li>
-                            For each type <fr:tex>A</fr:tex>, there is a distinguished term
-                            <fr:tex>[]</fr:tex> of type <fr:tex>LA</fr:tex>.
+                            For each type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a distinguished term
+                            <fr:tex
+display="inline"><![CDATA[[]]]></fr:tex> of type <fr:tex
+display="inline"><![CDATA[LA]]></fr:tex>.
                         </fr:li>
                         <fr:li>
-                            If <fr:tex>s : A</fr:tex> and <fr:tex>t : LA</fr:tex>, then <fr:tex>s::t : LA</fr:tex>.
+                            If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : LA]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s::t : LA]]></fr:tex>.
                         </fr:li></fr:ul></fr:li>
                 <fr:li>
                     (ix) The remaining term constructor associated with list
@@ -835,7 +1578,8 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
                     omit for irrelevance].
                 </fr:li></fr:ul></fr:p><fr:p>
             Of course, clauses (v) and (vi) of the definition above apply only
-            if we have function types in <fr:tex>\mathcal {L}</fr:tex>, clause (vii) only if we have power
+            if we have function types in <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>, clause (vii) only if we have power
             types, and clauses (viii) and (ix) only if we have list types.
             Note that, except for power types, the clauses come in pairs [of
             constructors and destructors]. The reason why we have no
@@ -843,139 +1587,684 @@ root="false"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:title>Terms</fr:title
             power types builds terms out of formulae rather than terms, the
             corresponding destructor builds formulae out of terms of a power
             type-- it thus appears as clause (iii) of the next definition."
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>252</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>263</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#183</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             (We again omit the bookkeeping of free variables.)
-            "The <fr:em>atomic formulae</fr:em> over <fr:tex>\mathcal {L}</fr:tex> are of three kinds:
+            "The <fr:em>atomic formulae</fr:em> over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are of three kinds:
             <fr:ul><fr:li>
-                    (i) If <fr:tex>R ↣ A</fr:tex> is a relation symbol and <fr:tex>t : A</fr:tex>, then
-                    <fr:tex>R(t)</fr:tex> is an atomic formulae.
+                    (i) If <fr:tex
+display="inline"><![CDATA[R ↣ A]]></fr:tex> is a relation symbol and <fr:tex
+display="inline"><![CDATA[t : A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[R(t)]]></fr:tex> is an atomic formulae.
                 </fr:li>
                 <fr:li>
-                    (ii) If <fr:tex>s</fr:tex> and <fr:tex>t</fr:tex> are terms of the same type <fr:tex>A</fr:tex>, then
-                    <fr:tex>s =_A t</fr:tex> is an atomic formula.
+                    (ii) If <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t]]></fr:tex> are terms of the same type <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then
+                    <fr:tex
+display="inline"><![CDATA[s =_A t]]></fr:tex> is an atomic formula.
                 </fr:li>
                 <fr:li>
-                    (iii) If <fr:tex>s : A</fr:tex> and <fr:tex>t :  \mathcal {P} A</fr:tex> for some <fr:tex>A</fr:tex>, then <fr:tex>s                     ∈_A t</fr:tex> is an atomic formula.
+                    (iii) If <fr:tex
+display="inline"><![CDATA[s : A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[t : \mathcal {P}A]]></fr:tex> for some <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, then <fr:tex
+display="inline"><![CDATA[s                     ∈_A t]]></fr:tex> is an atomic formula.
                 </fr:li></fr:ul></fr:p><fr:p>
-            The compound formulae over <fr:tex>\mathcal {L}</fr:tex> are built up from the atomic
-            formulae [with <fr:tex>⊤,∧,⊥,∨,⟹,¬,∃,∀,</fr:tex>]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            The compound formulae over <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex> are built up from the atomic
+            formulae [with <fr:tex
+display="inline"><![CDATA[⊤,∧,⊥,∨,⟹,¬,∃,∀,]]></fr:tex>]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>257</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Syntax à la <fr:link
-href="mclarty-1992.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>264</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title>Syntax à la <fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (McLarty defines the internal language of a topos <fr:tex>\bold { \mathsf {E}}</fr:tex> directly (and its
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 14.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (McLarty defines the internal language of a topos <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex> directly (and its
         interpretation), instead of describing the logic abstractly and taking
         the internal language to be a diagram (in Tarskian-model theory speak)
-        of <fr:tex>\bold { \mathsf {E}}</fr:tex>. We rephrase McLarty's presentation to not mention a particular
-        model <fr:tex>\bold { \mathsf {E}}</fr:tex>. We also preserve usage of the term "sort" instead of "type".)
+        of <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We rephrase McLarty's presentation to not mention a particular
+        model <fr:tex
+display="inline"><![CDATA[\bold {\mathsf {E}}]]></fr:tex>. We also preserve usage of the term "sort" instead of "type".)
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>254</fr:anchor><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>265</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title>Types</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             The types have the structure of the objects of an elementary topos,
-            ie a collection of atomic types closed under <fr:tex>1,×,→,Ω</fr:tex>.
-        </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            ie a collection of atomic types closed under <fr:tex
+display="inline"><![CDATA[1,×,→,Ω]]></fr:tex>.
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>255</fr:anchor><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title>Terms</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
             "The [language] is sorted…. We inductively define terms and their sorts:
             <fr:ul><fr:li>
-                    (LT1) Each sort <fr:tex>A</fr:tex> has a list of <fr:em>variables over <fr:tex>A</fr:tex></fr:em>,
-                    <fr:tex>x₁, x₂, x₃…</fr:tex>. Every variable over <fr:tex>A</fr:tex> is a term of sort
-                    <fr:tex>A</fr:tex>.
+                    (LT1) Each sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> has a list of <fr:em>variables over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex></fr:em>,
+                    <fr:tex
+display="inline"><![CDATA[x₁, x₂, x₃…]]></fr:tex>. Every variable over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is a term of sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT2) For any [function symbol <fr:tex>f : A→B</fr:tex>] and term <fr:tex>s</fr:tex> [of] sort
-                    <fr:tex>A</fr:tex>, <fr:tex>f s</fr:tex> is a term of sort <fr:tex>B</fr:tex>. [Constant symbols <fr:tex>c :                     1→A</fr:tex> are also] a term of sort <fr:tex>A</fr:tex>…. We write <fr:tex>!</fr:tex> as a
-                    constant [of sort <fr:tex>1</fr:tex>].
+                    (LT2) For any [function symbol <fr:tex
+display="inline"><![CDATA[f : A→B]]></fr:tex>] and term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> [of] sort
+                    <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[f s]]></fr:tex> is a term of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. [Constant symbols <fr:tex
+display="inline"><![CDATA[c :                     1→A]]></fr:tex> are also] a term of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>…. We write <fr:tex
+display="inline"><![CDATA[!]]></fr:tex> as a
+                    constant [of sort <fr:tex
+display="inline"><![CDATA[1]]></fr:tex>].
                 </fr:li>
                 <fr:li>
-                    (LT3) For every term <fr:tex>s₁</fr:tex> of sort <fr:tex>A</fr:tex> and <fr:tex>s₂</fr:tex> of sort <fr:tex>B</fr:tex>,
-                    there is a term <fr:tex>⟨s₁,s₂⟩</fr:tex> of sort <fr:tex>A×B</fr:tex>.
+                    (LT3) For every term <fr:tex
+display="inline"><![CDATA[s₁]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[s₂]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>,
+                    there is a term <fr:tex
+display="inline"><![CDATA[⟨s₁,s₂⟩]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[A×B]]></fr:tex>.
                 </fr:li>
                 <fr:li>
-                    (LT4) For every term <fr:tex>s</fr:tex> of sort <fr:tex>B</fr:tex> and variable <fr:tex>y</fr:tex> of
-                    sort <fr:tex>A</fr:tex>, there is a term <fr:tex>λy. s</fr:tex> of sort <fr:tex>Bᴬ</fr:tex>.
+                    (LT4) For every term <fr:tex
+display="inline"><![CDATA[s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> of
+                    sort <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, there is a term <fr:tex
+display="inline"><![CDATA[λy. s]]></fr:tex> of sort <fr:tex
+display="inline"><![CDATA[Bᴬ]]></fr:tex>.
                 </fr:li></fr:ul>"
             <fr:ul><fr:li>
-                    "The <fr:em>extension</fr:em> of <fr:tex>ϕ</fr:tex> over a list of variables
-                    <fr:tex>\overline {x}</fr:tex> is the sub-object <fr:tex>\{ \overline {x} | ϕ \}</fr:tex>
-                    of <fr:tex>A₁×A₂×…Aₙ</fr:tex> classified by [<fr:tex>ϕ</fr:tex>]."
-                </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    "The <fr:em>extension</fr:em> of <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> over a list of variables
+                    <fr:tex
+display="inline"><![CDATA[\overline {x}]]></fr:tex> is the sub-object <fr:tex
+display="inline"><![CDATA[\{\overline {x} | ϕ\}]]></fr:tex>
+                    of <fr:tex
+display="inline"><![CDATA[A₁×A₂×…Aₙ]]></fr:tex> classified by [<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>]."
+                </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>256</fr:anchor><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-            "A formula is a term of sort <fr:tex>Ω</fr:tex>. By LT2 [and the existence of
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title>Formulas</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>#187</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A formula is a term of sort <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>. By LT2 [and the existence of
             these terms in any elementary topos, developed in 13.3 and 13.4],
-            for any formulas <fr:tex>ϕ</fr:tex> and <fr:tex>ψ</fr:tex> there are formulas <fr:tex>ϕ∧ψ</fr:tex> and
-            <fr:tex>ϕ⟹ψ</fr:tex>.… There are also formulas <fr:tex>⊥, ¬ϕ, ϕ∧ψ</fr:tex>, read as 'false',
-            'not-<fr:tex>ϕ</fr:tex>', and '<fr:tex>ϕ</fr:tex> or <fr:tex>ψ</fr:tex>', which [with <fr:tex>∃</fr:tex> are defined in
-            terms of <fr:tex>∧,⟹,∀</fr:tex>] using quantification over <fr:tex>Ω</fr:tex> (15.1)."
+            for any formulas <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex> there are formulas <fr:tex
+display="inline"><![CDATA[ϕ∧ψ]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[ϕ⟹ψ]]></fr:tex>.… There are also formulas <fr:tex
+display="inline"><![CDATA[⊥, ¬ϕ, ϕ∧ψ]]></fr:tex>, read as 'false',
+            'not-<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex>', and '<fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> or <fr:tex
+display="inline"><![CDATA[ψ]]></fr:tex>', which [with <fr:tex
+display="inline"><![CDATA[∃]]></fr:tex> are defined in
+            terms of <fr:tex
+display="inline"><![CDATA[∧,⟹,∀]]></fr:tex>] using quantification over <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> (15.1)."
         </fr:p><fr:p>
-            "For any formula <fr:tex>ϕ</fr:tex> and variable <fr:tex>y</fr:tex> over any object <fr:tex>A</fr:tex>, we
-            define the formula <fr:tex>∀y : A. ϕ</fr:tex> as an abbreviation for <fr:tex>∀_A \{ y : A             | ϕ \}</fr:tex> [where <fr:tex>\{ y : A | ϕ \}</fr:tex> is an abbreviation for <fr:tex>λy : A. ϕ</fr:tex>,
-            and <fr:tex>∀ \_ A : Ω^A→Ω</fr:tex> is a term]."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            "For any formula <fr:tex
+display="inline"><![CDATA[ϕ]]></fr:tex> and variable <fr:tex
+display="inline"><![CDATA[y]]></fr:tex> over any object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>, we
+            define the formula <fr:tex
+display="inline"><![CDATA[∀y : A. ϕ]]></fr:tex> as an abbreviation for <fr:tex
+display="inline"><![CDATA[∀_A\{y : A             | ϕ\}]]></fr:tex> [where <fr:tex
+display="inline"><![CDATA[\{y : A | ϕ\}]]></fr:tex> is an abbreviation for <fr:tex
+display="inline"><![CDATA[λy : A. ϕ]]></fr:tex>,
+            and <fr:tex
+display="inline"><![CDATA[∀\_A : Ω^A→Ω]]></fr:tex> is a term]."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>258</fr:anchor><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>268</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title>Summary</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000M</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Ignoring the natural numbers type and the idiosyncratic list type of
         <fr:link
-href="johnstone-2002.xml"
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link>, there is a core concept (using the notion of
         elementary topops for reference) of <fr:link
-href="hejohns-000F.xml"
 type="local"
-addr="hejohns-000F">"finite-type intuitionistic
+href="hejohns-000F.xml"
+addr="hejohns-000F"
+title="&quot;Higher-order intuitionistic logic&quot; v. &quot;topos logic&quot;">"finite-type intuitionistic
         predicate logic" aka "topos logic" aka "higher-order intuitionistic
         logic"</fr:link>, although certain inter-definable choices can be
         made.
     </fr:p><fr:p>
         There's no clear notion of "taking everything", but a reasonable
-        "logical" choice seems to be that (over a language <fr:tex>\mathcal {L}</fr:tex>,) the types are
-        <fr:tex>→,×,1</fr:tex>, the terms are <fr:tex>x,λ,f t,⟨t,u⟩,πᵢ, \{ x|ϕ(x) \} ,*</fr:tex>, and the
-        formulas are <fr:tex>=,R( \overline {x}),∈,∧,∨,¬,⟹,∀,∃</fr:tex>.
+        "logical" choice seems to be that (over a language <fr:tex
+display="inline"><![CDATA[\mathcal {L}]]></fr:tex>,) the types are
+        <fr:tex
+display="inline"><![CDATA[→,×,1]]></fr:tex>, the terms are <fr:tex
+display="inline"><![CDATA[x,λ,f t,⟨t,u⟩,πᵢ,\{x|ϕ(x)\},*]]></fr:tex>, and the
+        formulas are <fr:tex
+display="inline"><![CDATA[=,R(\overline {x}),∈,∧,∨,¬,⟹,∀,∃]]></fr:tex>.
     </fr:p><fr:p>
-        One curiosity with taking a type <fr:tex>Ω</fr:tex> and formulas to <fr:em>just</fr:em> be
-        terms of type <fr:tex>Ω</fr:tex>, is that we then have formulas like <fr:tex>π₁ ⟨⊤,⊥⟩</fr:tex>
-        (definitionally equal to <fr:tex>⊤</fr:tex>) which don't seem very formula-like.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree>
+        One curiosity with taking a type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> and formulas to <fr:em>just</fr:em> be
+        terms of type <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, is that we then have formulas like <fr:tex
+display="inline"><![CDATA[π₁ ⟨⊤,⊥⟩]]></fr:tex>
+        (definitionally equal to <fr:tex
+display="inline"><![CDATA[⊤]]></fr:tex>) which don't seem very formula-like.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>307</fr:anchor><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>269</fr:anchor><fr:addr>#192</fr:addr><fr:route>unstable-192.xml</fr:route><fr:title>Semantics</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /><fr:parent>hejohns-000J</fr:parent></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>707</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    The usual 30-second description of an elementary topos, is that "it's like
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
+    Of course, these are always the kind of descriptions that only make sense
+    once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
+    We remark that <fr:link
+type="local"
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
+    perspective.
+</fr:p><fr:p>
+    Instead, I've gathered every book I could conveniently get my hands on just
+    now, and we'll see how each defines an elementary topos.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
+        classifier and a natural numbers object."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "We are now ready to present the definition of an (elementary) topos,
+        which is the central notion of this book: it is a category having all
+        the good 'set-like' properties discussed in the previous chapter,
+        except (possibly) for Booleanness. It is a remarkable fact that all
+        these properties are implied by the conjunction of just two of them:"
+    </fr:p><fr:p>
+        "A <fr:em>topos</fr:em> is a properly cartesian closed category with a subobject
+        classifier."
+    </fr:p><fr:p>
+        "… Before that, however, we must mention that the above definition of a
+        topos, though it is appreciably simplified from that originally adopted
+        by F. W. Lawvere and M. Tierney when they began the subject in 1969, is
+        not the simplest possible; one can combine the exponentials and the
+        subobject classifier into a single structure, that of power objects."
+    </fr:p><fr:p>
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        products and all its objects are exponentiable. … Nevertheless, most of
+        the cartesian closed categories we shall meet will have equalizers as
+        well as finite products; we shall indicate this (with a note of mild
+        reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
+        classifier."
+    </fr:p><fr:p>
+        Note though, that McLarty "assumes a base category with finite limits"
+        when talking about cartesian closed categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Okay so they only define a Grothendieck topos. We'll leave the
+        discussion of Grothendieck topoi to another time.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
+            </fr:li>
+            <fr:li>
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
+            </fr:li>
+            <fr:li>
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
+            </fr:li></fr:ul>
+        This compact definition proves to be amazingly rich in consequences:
+        it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
+        <fr:ul><fr:li>
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
+            </fr:li>
+            <fr:li>
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
+            </fr:li>
+            <fr:li>
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
+            </fr:li>
+            <fr:li>
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
+            </fr:li></fr:ul>
+        As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
+        <fr:ul><fr:li>
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
+            </fr:li></fr:ul>
+        and dually (2) replaced by
+        <fr:ul><fr:li>
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
+            </fr:li></fr:ul>
+        The definition just given is the one originally proposed by Lawvere and
+        Tierney, in terms of which they started [elementary] topos theory in
+        1969. Subsequently C. Juul Mikkelsen discovered that condition (2) is
+        implied by the combination of (1), (3) and (4) (cf. Paré [1974]). Thus
+        a topos can be defined as a Cartesian closed category with a subobject
+        classifier. In 4.7 we shall consider a different definition, based on a
+        categorial characterisation of power sets."
+    </fr:p><fr:p>
+        (<fr:em>complete</fr:em>, meaning every diagram has a limit, <fr:em>finitely
+        complete</fr:em>, meaning every finite diagram (finite objects, finite
+        morphisms) has a limit, "constitute", meaning that "cartesian closed"
+        is defined as (1) and (3), not just (3) as one expects nowadays.  See,
+        <fr:link
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
+            <fr:ul><fr:li>
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
+                </fr:li>
+                <fr:li>
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
+                </fr:li>
+                <fr:li>
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
+            classifier</fr:em>, as already described in §I.3, while in the second case
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
+            any category which satisfies (1) and has all exponentials will, in
+            particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
+type="local"
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
+            course of this chapter that the two sets of axioms are actually
+            equivalent.
+        </fr:p><fr:p>
+            Notice that the two natural isomorphisms (1) and (2) may be combined
+            in a single isomorphism
+            <fr:tex
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
+            also (2). In other words, we could define a topos as a category with
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
+    every definition, and give a standard one:
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
+type="local"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    So comparing each definition with another, they're all about par for
+    course, if we ignore natural numbers objects for now.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>709</fr:anchor><fr:addr>hejohns-000L</fr:addr><fr:route>hejohns-000L.xml</fr:route><fr:title>Generalized element</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A <fr:em>generalized element</fr:em> (alt. local element) of an object <fr:tex
+display="inline"><![CDATA[A ∈     \mathcal {C}]]></fr:tex> is <fr:em>just</fr:em> a morphism <fr:tex
+display="inline"><![CDATA[e : B → A]]></fr:tex>, for some <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>. Ie, a
+    generalized element of <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> is <fr:em>just</fr:em> an object of the slice over <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>.
+</fr:p><fr:p>
+    There's supposed to be some Kripke-style interpretation of these as
+    elements defined at points in time, but TODO.
+</fr:p><fr:p>
+    From <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
+type="local"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global element</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>710</fr:anchor><fr:addr>hejohns-000K</fr:addr><fr:route>hejohns-000K.xml</fr:route><fr:title>Global element</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    A <fr:em>global element</fr:em> of an object <fr:tex
+display="inline"><![CDATA[A ∈ \mathcal {C}]]></fr:tex> is <fr:em>just</fr:em> a
+    morphism <fr:tex
+display="inline"><![CDATA[e : 1 → A]]></fr:tex>.
+</fr:p><fr:p>
+    From <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>. Compare with <fr:link
+type="local"
+href="hejohns-000L.xml"
+addr="hejohns-000L"
+title="Generalized element">generalized element</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>711</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
+    <fr:embedded-tex
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
+    is a pullback square.
+</fr:p><fr:p>
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
+type="local"
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
+    element</fr:link> is trivially monic. Recall that "monos are stable
+    under pullback".
+</fr:p><fr:p>
+    This is from <fr:link
+type="local"
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/millies-zeilberger-2015.xml b/_site/forest/millies-zeilberger-2015.xml
index f1ce8b3..c2b437d 100644
--- a/_site/forest/millies-zeilberger-2015.xml
+++ b/_site/forest/millies-zeilberger-2015.xml
@@ -1,20 +1,20 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>592</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>722</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -30,18 +30,20 @@ root="false"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:taxon>Abstract</fr:ta
         present a few applications. As a larger case study, we revisit
         Reynolds’ paper on “The Meaning of Types” (2000), showing how the
         paper’s main results may be reconstructed along these lines.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
-        A LR is just a model of the syntax, used to prove <fr:tex>∀M</fr:tex> statements.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:em>What</fr:em> is a logical relation?
+        A LR is just a model of the syntax, used to prove <fr:tex
+display="inline"><![CDATA[∀M]]></fr:tex> statements.
         Thus, it's usually <fr:em>just</fr:em> a model "above" (or a structure preserving
         functor into) the syntax, typically trivially constructed as a pullback
         (it's hard in general to concoct models with structure preserving maps
@@ -93,21 +95,29 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>millies-z
     </fr:p><fr:p>
         Maybe I'm finally close to being able to understand what a Kripke
         logical relation is.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>590</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>723</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     What is the actual <fr:em>logic</fr:em> when doing a logical relation?
     Do you only actually need functors? Zeilberger's <fr:link
-href="millies-zeilberger-2015.pdf"
-type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
+type="external"
+href="millies-zeilberger-2015.pdf">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>must read</fr:strong>.
     Then you lose the internal logic... maybe? Maybe you just have a different internal logic?
 </fr:p><fr:p>
     Talked to Max about this on 5-9.
@@ -115,6 +125,7 @@ type="external">Functors as Type Refinement Systems</fr:link>, is a <fr:strong>m
     a hoare logic, and a implicational (eg FO) logic.
     I suppose one'd want to study a bit the relationship between the two.
     <fr:link
-href="millies-zeilberger-2015.xml"
 type="local"
-addr="millies-zeilberger-2015">paper</fr:link></fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="millies-zeilberger-2015.xml"
+addr="millies-zeilberger-2015"
+title="Functors are Type Refinement Systems">paper</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/prelude.xml b/_site/forest/prelude.xml
index 266e64c..7badef5 100644
--- a/_site/forest/prelude.xml
+++ b/_site/forest/prelude.xml
@@ -1,10 +1,10 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>823</fr:anchor><fr:addr>prelude</fr:addr><fr:route>prelude.xml</fr:route></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>934</fr:anchor><fr:addr>prelude</fr:addr><fr:route>prelude.xml</fr:route><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/quine-1960.xml b/_site/forest/quine-1960.xml
index a6c734b..2725ea4 100644
--- a/_site/forest/quine-1960.xml
+++ b/_site/forest/quine-1960.xml
@@ -1,13 +1,13 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>564</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>751</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
     this list. Whatever else you might think about Quine, he was a terrific
     explainer, and this is the best introduction to the subject I know."
@@ -15,16 +15,17 @@ root="false"><fr:frontmatter><fr:anchor>564</fr:anchor><fr:taxon>Reference</fr:t
     (Yes, that 50 should be a 60(b).)
     Here is the followup I wrote:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Thanks for sharing this nice little note, which sketches a combinatory
         (classical) first-order logic.
     </fr:p><fr:p>
@@ -47,8 +48,8 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-196
         wants to recover from any definite property ("logical condition") on
         I-objects, the (existence of a) class (II-object) of those I-objects.
         Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
+type="external"
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem">class existence theorem</fr:link>.
     </fr:p><fr:p>
         (More generally, the "reduction theorem" in von Neumann 1925 says that
         every expression of I-objects (definable class-function) is indeed a
@@ -74,46 +75,65 @@ type="external">class existence theorem</fr:link>.
         nontrivial, and found this connection with Quine's note interesting,
     </fr:p><fr:p>
         Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:ul><fr:li><fr:link
+type="external"
+href="quine-1960.pdf">quine-1960.pdf</fr:link></fr:li>
     <fr:li><fr:link
-href="von-neumann-1925.xml"
 type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>752</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
-href="quine-1960.xml"
 type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
+href="quine-1960.xml"
+addr="quine-1960"
+title="Variables Explained Away">Variables Explained Away</fr:link>.
 </fr:p><fr:p>
     English translation in <fr:link
-href="van-heijenoort-1967.xml"
 type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="van-heijenoort-1967.xml"
+addr="van-heijenoort-1967"
+title="From Frege To Gödel">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>562</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>753</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
-href="quine-1960.xml"
 type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
+href="quine-1960.xml"
+addr="quine-1960"
+title="Variables Explained Away">Variables Explained Away</fr:link>.
 </fr:p><fr:p>
     English translation in <fr:link
-href="van-heijenoort-1967.xml"
 type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="van-heijenoort-1967.xml"
+addr="van-heijenoort-1967"
+title="From Frege To Gödel">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/ronnie-chen.xml b/_site/forest/ronnie-chen.xml
index aa99450..77063c5 100644
--- a/_site/forest/ronnie-chen.xml
+++ b/_site/forest/ronnie-chen.xml
@@ -1,25 +1,33 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>500</fr:anchor><fr:taxon>Person</fr:taxon><fr:addr>ronnie-chen</fr:addr><fr:route>ronnie-chen.xml</fr:route><fr:title>Ronnie Chen</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>650</fr:anchor><fr:addr>ronnie-chen</fr:addr><fr:route>ronnie-chen.xml</fr:route><fr:title>Ronnie Chen</fr:title><fr:taxon>Person</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>499</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>651</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
-href="ronnie-chen.xml"
 type="local"
-addr="ronnie-chen">Ronnie Chen</fr:link>,
+href="ronnie-chen.xml"
+addr="ronnie-chen"
+title="Ronnie Chen">Ronnie Chen</fr:link>,
     where the viewpoint of "generalized algebraic semantics" is developed.
 </fr:p><fr:p><fr:link
-href="marquis-reyes-2012.pdf"
-type="external">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+type="external"
+href="marquis-reyes-2012.pdf">marquis-reyes-2012.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/scherer-2017.xml b/_site/forest/scherer-2017.xml
index e217fd3..e68ec1d 100644
--- a/_site/forest/scherer-2017.xml
+++ b/_site/forest/scherer-2017.xml
@@ -1,40 +1,49 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>575</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>698</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
+type="local"
 href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">Equality Between Functionals</fr:link> and <fr:link
 type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
 href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
+addr="statman-1985"
+title="Equality Between Functionals Revisited">Equality Between Functionals Revisited</fr:link>.
     Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://proofassistants.stackexchange.com/a/1177">stack overflow</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>573</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>699</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -51,37 +60,39 @@ root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:ta
         Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
         with or without types, have been considered. The motivation for introducing
         these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex
+display="inline"><![CDATA[⊢ s = t]]></fr:tex> defined
         in the text, is equivalent to one of these definitions of convertibility.
         It is easy to see that any two convertible terms define the same functional
         of finite type. We show here that any two non- convertible terms define
         different functionals of finite type.[This result was obtained in 1970.]
         This, coupled with the known decidability of convertibility, tells us that
         "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     Square brackets denote author's footnotes.
     <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="friedman-1975.pdf">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>574</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>700</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
-href="friedman-1975.xml"
 type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">3</fr:link>) in the context of what we know today about the model
         theory of the typed λ-calculus. In order to do this, it is appropriate to
         survey recent work in this area, in so far as it touches on issues raised
         by Friedman. We don't intend a general survey; much will be left out. In
@@ -104,6 +115,6 @@ addr="friedman-1975">3</fr:link>) in the context of what we know today about the
             </fr:li>
             <fr:li>
                 logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+            </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="statman-1985.pdf">statman-1985.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/shoenfield-1967.xml b/_site/forest/shoenfield-1967.xml
index 86f1b05..b146e24 100644
--- a/_site/forest/shoenfield-1967.xml
+++ b/_site/forest/shoenfield-1967.xml
@@ -1,32 +1,40 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>267</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>523</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>266</fr:anchor><fr:addr>logicmatters</fr:addr><fr:route>logicmatters.xml</fr:route><fr:title>Peter Smith's Book Reviews</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>524</fr:anchor><fr:addr>logicmatters</fr:addr><fr:route>logicmatters.xml</fr:route><fr:title>Peter Smith's Book Reviews</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of my all-time favorite pages on the web (and that's not an uncommonly
     held opinion), a tremendous resource and service for the community. I'm not
     able to evaluate too many of Prof. Smith's opinions, but I don't disagree
     with any of what he's written about the books I have seen.
 </fr:p><fr:p>
     At least I just reread the review of <fr:link
-href="shoenfield-1967.xml"
 type="local"
-addr="shoenfield-1967">Mathematical Logic</fr:link>, and "So this
+href="shoenfield-1967.xml"
+addr="shoenfield-1967"
+title="Mathematical Logic">Mathematical Logic</fr:link>, and "So this
     neither has the cleanness of a Hilbert system nor the naturalness of a
     natural deduction system. As far as I noticed, nothing is said to motivate
     this seemingly horrible choice as against others." always makes me laugh.
 </fr:p><fr:p><fr:link
-href="https://www.logicmatters.net/tyl/booknotes/"
-type="external">https://www.logicmatters.net/tyl/booknotes/</fr:link></fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+type="external"
+href="https://www.logicmatters.net/tyl/booknotes/">https://www.logicmatters.net/tyl/booknotes/</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/statman-1985.xml b/_site/forest/statman-1985.xml
index 465a561..80846b3 100644
--- a/_site/forest/statman-1985.xml
+++ b/_site/forest/statman-1985.xml
@@ -1,25 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>581</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>715</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
-href="friedman-1975.xml"
 type="local"
-addr="friedman-1975">3</fr:link>) in the context of what we know today about the model
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">3</fr:link>) in the context of what we know today about the model
         theory of the typed λ-calculus. In order to do this, it is appropriate to
         survey recent work in this area, in so far as it touches on issues raised
         by Friedman. We don't intend a general survey; much will be left out. In
@@ -42,43 +43,30 @@ addr="friedman-1975">3</fr:link>) in the context of what we know today about the
             </fr:li>
             <fr:li>
                 logical relations.
-            </fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:p><fr:link
-href="statman-1985.pdf"
-type="external">statman-1985.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+            </fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p><fr:link
+type="external"
+href="statman-1985.pdf">statman-1985.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>580</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="https://arxiv.org/pdf/1610.01213"
-type="external">https://arxiv.org/pdf/1610.01213</fr:link>.
-    The nice introduction informed me of <fr:link
-href="friedman-1975.xml"
-type="local"
-addr="friedman-1975">Equality Between Functionals</fr:link> and <fr:link
-href="statman-1985.xml"
-type="local"
-addr="statman-1985">Equality Between Functionals Revisited</fr:link>.
-    Referenced on <fr:link
-href="https://proofassistants.stackexchange.com/a/1177"
-type="external">stack overflow</fr:link>.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>579</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>716</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -95,15 +83,45 @@ root="false"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:taxon>Abstract</fr:ta
         Several definitions of <fr:em>convertibility</fr:em> between terms in the λ-calculus,
         with or without types, have been considered. The motivation for introducing
         these definitions has primarily been to analyze the notion of the
-        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex>⊢ s = t</fr:tex> defined
+        <fr:em>identity</fr:em> between rules, or processes. The relation <fr:tex
+display="inline"><![CDATA[⊢ s = t]]></fr:tex> defined
         in the text, is equivalent to one of these definitions of convertibility.
         It is easy to see that any two convertible terms define the same functional
         of finite type. We show here that any two non- convertible terms define
         different functionals of finite type.[This result was obtained in 1970.]
         This, coupled with the known decidability of convertibility, tells us that
         "equality between simple functionals is recursive."
-    </fr:p></fr:mainmatter></fr:tree><fr:p>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     Square brackets denote author's footnotes.
     <fr:link
-href="friedman-1975.pdf"
-type="external">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+type="external"
+href="friedman-1975.pdf">friedman-1975.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>717</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+type="external"
+href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
+    The nice introduction informed me of <fr:link
+type="local"
+href="friedman-1975.xml"
+addr="friedman-1975"
+title="Equality Between Functionals">Equality Between Functionals</fr:link> and <fr:link
+type="local"
+href="statman-1985.xml"
+addr="statman-1985"
+title="Equality Between Functionals Revisited">Equality Between Functionals Revisited</fr:link>.
+    Referenced on <fr:link
+type="external"
+href="https://proofassistants.stackexchange.com/a/1177">stack overflow</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/sterling-2022.xml b/_site/forest/sterling-2022.xml
index bab8300..e9807de 100644
--- a/_site/forest/sterling-2022.xml
+++ b/_site/forest/sterling-2022.xml
@@ -1,20 +1,20 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>571</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>728</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -28,14 +28,14 @@ root="false"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:taxon>Abstract</fr:ta
         computability</fr:strong>, logical relations are manipulated as if they were sets,
         making the essence of many complex logical relations arguments
         accessible to non-specialists.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -48,32 +48,40 @@ root="false"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:date><fr:year>2024</f
         and I just don't have some context, or it's actually presented almost
         mystically.
         The mention of <fr:link
-href="carette-kiselyov-shan-2007.xml"
 type="local"
-addr="carette-kiselyov-shan-2007">finally tagless encodings</fr:link>
+href="carette-kiselyov-shan-2007.xml"
+addr="carette-kiselyov-shan-2007"
+title="Finally Tagless, Partially Evaluated">finally tagless encodings</fr:link>
         and <fr:link
-href="hoffman-1999"
-type="external">a justification for hoas</fr:link> doesn't seem super
+type="external"
+href="hoffman-1999">a justification for hoas</fr:link> doesn't seem super
         on-the-point, but if it's mentioned it must be for good reason…
     </fr:p><fr:p>
         Again, quotes are air-quotes, not actual quotes.
         <fr:link
-href="sterling-2022.pdf"
-type="external">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="sterling-2022.pdf">sterling-2022.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>570</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>729</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -94,14 +102,14 @@ root="false"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:taxon>Abstract</fr:ta
         stuck.  To achieve self-interpretation and show Jones-optimality, we
         relate this exemplar of higher-rank and higher-kind polymorphism to
         plugging a term into a context of let-polymorphic bindings.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -123,4 +131,4 @@ root="false"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:taxon>Notes</fr:taxon
         that it is the free syntax generated by the encoded rules/initial
         algebra/inductive dataype.
         But the naïve TODO
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/street-2012.xml b/_site/forest/street-2012.xml
index aa53be9..e28d18a 100644
--- a/_site/forest/street-2012.xml
+++ b/_site/forest/street-2012.xml
@@ -1,22 +1,22 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>703</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>758</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -25,568 +25,429 @@ root="false"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:taxon>Abstract</fr:ta
         only on objects and the usual hom adjointness isomorphism implied the
         effect of the functors on homs was uniquely determined. Writing this down
         properly led to the present paper.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>697</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
-        <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
-            <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
-        <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
-        commutes.
-        The adjunction is unique when it exists.
-    </fr:p><fr:p>
-        The <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
-        of Homset-categories here.
-    </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
-        <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
-        explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
-        circularity.
-    </fr:p><fr:p>
-        This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
-type="local"
-addr="street-2012">the
-        paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-        precursor</fr:link> head-on. The utility is that it holds in
-        enriched categories.
-    </fr:p><fr:p>
-        The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-        Then <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
-        true of an adjunction.
-        <fr:link
-href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
-    </fr:p><fr:p>
-        The nontrivial directions are the backward directions.
-    </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
-            </fr:p>
-            
-        <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-
-            <fr:p>
-                As does this trivial square:
-            </fr:p>
-            <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                which when pasted together, tells us magically:
-            </fr:p>
-            <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            <fr:p>
-                Combined with <fr:link
-href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
-                Actually, we saw earlier that <fr:link
-href="subb-000E.xml"
-type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
-            </fr:p>
-            <fr:p>
-                There's two ways ways to see the uniqueness situation:
-            </fr:p>
-            <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
-                    We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
-                    <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
-                </fr:li>
-                <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
-                    the unique one extending the core by applying the
-                    uniqueness result of <fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
-                    functor in turn, just knowing the other is a well-defined
-                    functor.
-                </fr:li></fr:ul></fr:li>
-        <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
-                This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
-type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>699</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>759</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Mentioned in passing as <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
         <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
     This is conventionally left implicit, but we remark to clarify what we mean
     by the family of natural isomorphisms being natural.
 </fr:p><fr:p>
     NOTE: I couldn't get よ to render in the latex (diagrams).
     I'm not sure if it's a lualatex/dvi or font issue.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -594,90 +455,436 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
     so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>761</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>701</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-    This is mentioned in passing, in the intro to <fr:link
 href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
+        <fr:ul><fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
+            <fr:li>
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
+        <fr:link
+type="local"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
+type="local"
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
+        commutes.
+        The adjunction is unique when it exists.
+    </fr:p><fr:p>
+        The <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
+        of Homset-categories here.
+    </fr:p><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
+        <fr:embedded-tex
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
+type="local"
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the
+        explicit proof of the precursor lemma</fr:link>.
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
+type="local"
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
+        circularity.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        This proof is rather low tech and is lifted straight from <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
+        paper</fr:link>. It's also how you'd attack <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+        precursor</fr:link> head-on. The utility is that it holds in
+        enriched categories.
+    </fr:p><fr:p>
+        The forward directions are trivial.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
+        Then <fr:link
+type="local"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
+        true of an adjunction.
+        <fr:link
+type="local"
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
+    </fr:p><fr:p>
+        The nontrivial directions are the backward directions.
+    </fr:p><fr:ul><fr:li><fr:p>
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
+            </fr:p>
+            
+        <fr:embedded-tex
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+
+            <fr:p>
+                As does this trivial square:
+            </fr:p>
+            <fr:embedded-tex
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                which when pasted together, tells us magically:
+            </fr:p>
+            <fr:embedded-tex
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            <fr:p>
+                Combined with <fr:link
+type="local"
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
+                Actually, we saw earlier that <fr:link
+type="local"
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                precursor lemma</fr:link>, thus there is a unique
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
+            </fr:p>
+            <fr:p>
+                There's two ways ways to see the uniqueness situation:
+            </fr:p>
+            <fr:ul><fr:li>
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+                    We've essentially seen this already, if not literally.
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
+                    <fr:embedded-tex
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
+                </fr:li>
+                <fr:li>
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
+                    the unique one extending the core by applying the
+                    uniqueness result of <fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
+                    functor in turn, just knowing the other is a well-defined
+                    functor.
+                </fr:li></fr:ul></fr:li>
+        <fr:li><fr:p>
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
+                This is dual.
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="subb-0006.xml"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>763</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+    This is mentioned in passing, in the intro to <fr:link
+type="local"
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
+</fr:p><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p><fr:p>
+        (dual)
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/sub-0006.xml b/_site/forest/sub-0006.xml
index 56f1e11..4c0058e 100644
--- a/_site/forest/sub-0006.xml
+++ b/_site/forest/sub-0006.xml
@@ -1,123 +1,175 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>666</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>772</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>664</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>774</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>662</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>776</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/sub-0007.xml b/_site/forest/sub-0007.xml
index 71fad7c..4f769a7 100644
--- a/_site/forest/sub-0007.xml
+++ b/_site/forest/sub-0007.xml
@@ -1,376 +1,500 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>721</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>891</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+type="local"
 href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>719</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>893</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     Mentioned in passing as <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">2.1. Proposition.</fr:link>, as "well-known".
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">2.1. Proposition.</fr:link>, as "well-known".
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose <fr:tex>U : A → X</fr:tex> is a functor, <fr:tex>F : X .ob → A .ob</fr:tex> a function,
-            and <fr:tex>(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}</fr:tex> a family of natural
-            isomorphisms. Then there is a unique extension <fr:tex>\overline {F}</fr:tex> of <fr:tex>F</fr:tex> such
-            that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is natural in <fr:tex>x</fr:tex>, witnessing <fr:tex>\overline {F}  ⊣ U</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose <fr:tex
+display="inline"><![CDATA[U : A → X]]></fr:tex> is a functor, <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex> a function,
+            and <fr:tex
+display="inline"><![CDATA[(πₓ : A(F(x), -) ≅ X(x, U(-)))_{x ∈ X}]]></fr:tex> a family of natural
+            isomorphisms. Then there is a unique extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> such
+            that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[x]]></fr:tex>, witnessing <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ U]]></fr:tex>.
         </fr:li>
         <fr:li>
-            Or dually, suppose that <fr:tex>F</fr:tex> is a functor, but <fr:tex>U</fr:tex> is just a
-            function on objects, and <fr:tex>(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}</fr:tex>.
-            Then <fr:tex>U</fr:tex> uniquely extends to <fr:tex>\overline {U}</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:p>
-    Let's be pedantic and clarify that <fr:tex>A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ</fr:tex>
-    have the same action on morphisms as <fr:tex>Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U</fr:tex>.
+            Or dually, suppose that <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> is a functor, but <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is just a
+            function on objects, and <fr:tex
+display="inline"><![CDATA[(πₐ : A(F(-), a) ≅ X(-, U(a)))_{a ∈ A}]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> uniquely extends to <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
+    Let's be pedantic and clarify that <fr:tex
+display="inline"><![CDATA[A(F(-), -), X(-, U(-)): Xᵒᵖ → Setᴬ]]></fr:tex>
+    have the same action on morphisms as <fr:tex
+display="inline"><![CDATA[Hom(-, -)∘Fᵒᵖ, λx. Hom(x, -)∘U]]></fr:tex>.
     This is conventionally left implicit, but we remark to clarify what we mean
     by the family of natural isomorphisms being natural.
 </fr:p><fr:p>
     NOTE: I couldn't get よ to render in the latex (diagrams).
     I'm not sure if it's a lualatex/dvi or font issue.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -378,287 +502,370 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     However, unlike <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>,
-    we crucially use the information that <fr:tex>U</fr:tex> is a functor
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>,
+    we crucially use the information that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor
     so the natural transformation squares are well-defined.
-</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>715</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>895</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Of course in hindsight, we could've (or even should've)
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
+        more clearly well-behaved definition, but unless you're
+        already sold on the Yoneda principle, I find it a difficult
+        leap without struggling with lower-level objects a bit
+        first.
+        I'm still in training.
+    </fr:p><fr:p>
+        This is the perspective taken in <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>897</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>717</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
-        more clearly well-behaved definition, but unless you're
-        already sold on the Yoneda principle, I find it a difficult
-        leap without struggling with lower-level objects a bit
-        first.
-        I'm still in training.
-    </fr:p><fr:p>
-        This is the perspective taken in <fr:link
-href="sub-0007.xml"
-type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/sub-000D.xml b/_site/forest/sub-000D.xml
index f867954..d9a5adc 100644
--- a/_site/forest/sub-000D.xml
+++ b/_site/forest/sub-000D.xml
@@ -1,236 +1,316 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>624</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>778</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>622</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>780</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -238,224 +318,304 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>620</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>782</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -463,7 +623,8 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/sub-000E.xml b/_site/forest/sub-000E.xml
index d952ea4..beceba8 100644
--- a/_site/forest/sub-000E.xml
+++ b/_site/forest/sub-000E.xml
@@ -1,296 +1,390 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>588</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>784</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>586</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>786</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>584</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>788</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/sub-000G.xml b/_site/forest/sub-000G.xml
index 2f6a53c..f1ddd10 100644
--- a/_site/forest/sub-000G.xml
+++ b/_site/forest/sub-000G.xml
@@ -1,81 +1,130 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>518</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>560</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title><fr:link
+type="local"
+href="unstable-199.xml"
+addr="#199"
+title="What is an elementary topos? › Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory"><fr:link
+type="local"
+href="hejohns-000G.xml"
+addr="hejohns-000G"
+title="What is an elementary topos?">What is an elementary topos?</fr:link> › Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:link> › Topos à la <fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>References</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>562</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>516</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>563</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -91,107 +140,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -206,177 +278,238 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>514</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>565</fr:anchor><fr:addr>hejohns-000G</fr:addr><fr:route>hejohns-000G.xml</fr:route><fr:title>What is an elementary topos?</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     The usual 30-second description of an elementary topos, is that "it's like
-    <fr:tex>Set</fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
-    of <fr:tex>Set</fr:tex>".
+    <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>" (but not quite), or "it's an axiomatization of the basic properties
+    of <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>".
     Of course, these are always the kind of descriptions that only make sense
     once you've <fr:em>already</fr:em> learned what a(n elementary) topos is.
     We remark that <fr:link
-href="mclarty-1990b.xml"
 type="local"
-addr="mclarty-1990b">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex>Set</fr:tex>"-based
+href="mclarty-1990b.xml"
+addr="mclarty-1990b"
+title="The Uses and Abuses of the History of Topos Theory">The Uses and Abuses of the History of Topos Theory</fr:link> is precisely about this "<fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>"-based
     perspective.
 </fr:p><fr:p>
     Instead, I've gathered every book I could conveniently get my hands on just
     now, and we'll see how each defines an elementary topos.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>283</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>233</fr:anchor><fr:addr>#193</fr:addr><fr:route>unstable-193.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="lambek-scott-1986.xml"
+addr="lambek-scott-1986"
+title="Introduction to higher order categorical logic">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="lambek-scott-1986">Introduction to higher order categorical logic</fr:link> p.140</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a cartesian closed category with a sub-object
         classifier and a natural numbers object."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>285</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>235</fr:anchor><fr:addr>#194</fr:addr><fr:route>unstable-194.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="johnstone-2002.xml"
+addr="johnstone-2002"
+title="Sketches of an Elephant: a topos theory compendium">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="johnstone-2002">Sketches of an Elephant: a topos theory compendium</fr:link> 2.1.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "We are now ready to present the definition of an (elementary) topos,
         which is the central notion of this book: it is a category having all
         the good 'set-like' properties discussed in the previous chapter,
@@ -392,107 +525,130 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         not the simplest possible; one can combine the exponentials and the
         subobject classifier into a single structure, that of power objects."
     </fr:p><fr:p>
-        ("We say a category <fr:tex>\mathcal {C}</fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
+        ("We say a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is <fr:em>cartesian closed</fr:em> if it has finite
         products and all its objects are exponentiable. … Nevertheless, most of
         the cartesian closed categories we shall meet will have equalizers as
         well as finite products; we shall indicate this (with a note of mild
         reproof in our voice) by calling them <fr:em>properly cartesian closed</fr:em>.")
-        (Recall that finite products <fr:tex>+</fr:tex> equalizers <fr:tex>⟺</fr:tex> finite limits.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        (Recall that finite products <fr:tex
+display="inline"><![CDATA[+]]></fr:tex> equalizers <fr:tex
+display="inline"><![CDATA[⟺]]></fr:tex> finite limits.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>287</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>237</fr:anchor><fr:addr>#195</fr:addr><fr:route>unstable-195.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link> 13.1</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         "A <fr:em>topos</fr:em> is a Cartesian closed category with a sub-object
         classifier."
     </fr:p><fr:p>
         Note though, that McLarty "assumes a base category with finite limits"
         when talking about cartesian closed categories.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>289</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>239</fr:anchor><fr:addr>#196</fr:addr><fr:route>unstable-196.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="makkai-reyes-1977.xml"
+addr="makkai-reyes-1977"
+title="First Order Categorical Logic">First Order Categorical Logic</fr:link></fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="makkai-reyes-1977">First Order Categorical Logic</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Okay so they only define a Grothendieck topos. We'll leave the
         discussion of Grothendieck topoi to another time.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>291</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>241</fr:anchor><fr:addr>#197</fr:addr><fr:route>unstable-197.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="awodey-2010.xml"
+addr="awodey-2010"
+title="Category Theory">Category Theory</fr:link> 8.17</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="awodey-2010">Category Theory</fr:link> 8.17</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "A <fr:em>topos</fr:em> is a category <fr:tex>\mathcal {E}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "A <fr:em>topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> such that
         <fr:ul><fr:li>
-                1. <fr:tex>\mathcal {E}</fr:tex> has all finite limits,
+                1. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits,
             </fr:li>
             <fr:li>
-                2. <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier,
+                2. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier,
             </fr:li>
             <fr:li>
-                3. <fr:tex>\mathcal {E}</fr:tex> has all exponentials.
+                3. <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all exponentials.
             </fr:li></fr:ul>
         This compact definition proves to be amazingly rich in consequences:
         it can be shown for instance that topoi also have all finite colimits, and that every slice category of a topos is again a topos."
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>293</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>243</fr:anchor><fr:addr>#198</fr:addr><fr:route>unstable-198.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="goldblatt-2006.xml"
+addr="goldblatt-2006"
+title="Topoi: The Categorial Analysis of Logic">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="goldblatt-2006">Topoi: The Categorial Analysis of Logic</fr:link> 4.3</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        "An <fr:em>elementary topos</fr:em> is a category <fr:tex>\mathcal {C}</fr:tex> such that
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        "An <fr:em>elementary topos</fr:em> is a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> such that
         <fr:ul><fr:li>
-                (1) <fr:tex>\mathcal {C}</fr:tex> is finitely complete,
+                (1) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely complete,
             </fr:li>
             <fr:li>
-                (2) <fr:tex>\mathcal {C}</fr:tex> is finitely co-complete,
+                (2) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> is finitely co-complete,
             </fr:li>
             <fr:li>
-                (3) <fr:tex>\mathcal {C}</fr:tex> has exponentiation,
+                (3) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has exponentiation,
             </fr:li>
             <fr:li>
-                (4) <fr:tex>\mathcal {C}</fr:tex> has a subobject classifier.
+                (4) <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a subobject classifier.
             </fr:li></fr:ul>
         As observed in Chapter 3, (1) and (3) constitute the definition of 'Cartesian closed' while (1) can be replaced by
         <fr:ul><fr:li>
-                (1') <fr:tex>\mathcal {C}</fr:tex> has a terminal object and pullbacks,
+                (1') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has a terminal object and pullbacks,
             </fr:li></fr:ul>
         and dually (2) replaced by
         <fr:ul><fr:li>
-                (2') <fr:tex>\mathcal {C}</fr:tex> has an initial object <fr:tex>0</fr:tex>, and pushouts.
+                (2') <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex> has an initial object <fr:tex
+display="inline"><![CDATA[0]]></fr:tex>, and pushouts.
             </fr:li></fr:ul>
         The definition just given is the one originally proposed by Lawvere and
         Tierney, in terms of which they started [elementary] topos theory in
@@ -507,128 +663,166 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         morphisms) has a limit, "constitute", meaning that "cartesian closed"
         is defined as (1) and (3), not just (3) as one expects nowadays.  See,
         <fr:link
-href="https://math.stackexchange.com/q/1845697"
-type="external">this stack overflow post</fr:link>.)
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+type="external"
+href="https://math.stackexchange.com/q/1845697">this stack overflow post</fr:link>.)
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>245</fr:anchor><fr:addr>#199</fr:addr><fr:route>unstable-199.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link></fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000G</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>247</fr:anchor><fr:addr>sub-000G</fr:addr><fr:route>sub-000G.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 48</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "[A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category] with the following properties
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "[A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category] with the following properties
             <fr:ul><fr:li>
-                    (i) <fr:tex>\mathcal {E}</fr:tex> has all finite limits and colimits,
+                    (i) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has all finite limits and colimits,
                 </fr:li>
                 <fr:li>
-                    (ii) <fr:tex>\mathcal {E}</fr:tex> has exponentials,
+                    (ii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has exponentials,
                 </fr:li>
                 <fr:li>
-                    (iii) <fr:tex>\mathcal {E}</fr:tex> has a subobject classifier <fr:tex>1 → Ω</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                    (iii) <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> has a subobject classifier <fr:tex
+display="inline"><![CDATA[1 → Ω]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:taxon>Definition</fr:taxon><fr:title>Topos à la <fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>249</fr:anchor><fr:addr>#200</fr:addr><fr:route>unstable-200.xml</fr:route><fr:title>Topos à la <fr:link
+type="local"
 href="maclane-moerdijk-1994.xml"
+addr="maclane-moerdijk-1994"
+title="Sheaves in Geometry and Logic: A First Introduction to Topos Theory">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="maclane-moerdijk-1994">Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:link> p. 161</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
-            "A <fr:em>topos</fr:em> <fr:tex>\mathcal {E}</fr:tex> is a category with all finite limites, equipped with
-            an object <fr:tex>Ω</fr:tex>, with a function <fr:tex>P</fr:tex> which assigns to each object <fr:tex>B</fr:tex>
-            of <fr:tex>\mathcal {E}</fr:tex> an object <fr:tex>PB</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, and, for each object <fr:tex>A</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, with two
-            isomorphisms, each natural in <fr:tex>A</fr:tex>
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#199</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+            "A <fr:em>topos</fr:em> <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> is a category with all finite limites, equipped with
+            an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex>, with a function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> which assigns to each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>
+            of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex> an object <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, and, for each object <fr:tex
+display="inline"><![CDATA[A]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, with two
+            isomorphisms, each natural in <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} } &amp;≅ Hom_{ \mathcal {E} }(A, Ω) &amp;(1) \\                      Hom_{ \mathcal {E} }(B × A, Ω) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(2)  \\                   \end {align*}             </fr:tex>
-            In other words, the functors <fr:tex>Sub_{ \mathcal {E} }</fr:tex> and <fr:tex>Hom_{ \mathcal {E} }(B × -, Ω)</fr:tex>, the
-            latter for each object <fr:tex>B</fr:tex> of <fr:tex>\mathcal {E}</fr:tex>, are required to be representable.
-            In the first case, the representing object <fr:tex>Ω</fr:tex> is the <fr:em>subobject
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}} &≅ Hom_{\mathcal {E}}(A, Ω) &(1)\\                     Hom_{\mathcal {E}}(B × A, Ω) &≅ Hom_{\mathcal {E}}(A, PB) &(2) \\                 \end {align*}             ]]></fr:tex>
+            In other words, the functors <fr:tex
+display="inline"><![CDATA[Sub_{\mathcal {E}}]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Hom_{\mathcal {E}}(B × -, Ω)]]></fr:tex>, the
+            latter for each object <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> of <fr:tex
+display="inline"><![CDATA[\mathcal {E}]]></fr:tex>, are required to be representable.
+            In the first case, the representing object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> is the <fr:em>subobject
             classifier</fr:em>, as already described in §I.3, while in the second case
-            <fr:tex>PB</fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex>B</fr:tex>; moreover, the
-            function <fr:tex>P</fr:tex> can be extended to a functor <fr:tex>P :  \mathcal {E} ^{op} →  \mathcal {E}</fr:tex> in
-            exactly one way so that (2) becomes natural in <fr:tex>B</fr:tex> (as well as in
-            <fr:tex>A</fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
-            that <fr:tex>PB</fr:tex> is the exponential <fr:tex>Ω^B</fr:tex>, as described in §I.6; therefore,
+            <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> may be called the <fr:em>power object</fr:em> of <fr:tex
+display="inline"><![CDATA[B]]></fr:tex>; moreover, the
+            function <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> can be extended to a functor <fr:tex
+display="inline"><![CDATA[P : \mathcal {E}^{op} → \mathcal {E}]]></fr:tex> in
+            exactly one way so that (2) becomes natural in <fr:tex
+display="inline"><![CDATA[B]]></fr:tex> (as well as in
+            <fr:tex
+display="inline"><![CDATA[A]]></fr:tex>), see (7) below. Equivalently, the natural isomorphism (2) states
+            that <fr:tex
+display="inline"><![CDATA[PB]]></fr:tex> is the exponential <fr:tex
+display="inline"><![CDATA[Ω^B]]></fr:tex>, as described in §I.6; therefore,
             any category which satisfies (1) and has all exponentials will, in
             particular, have all power objects. We emphasize that at first sight the axioms for at topos just giben seem weaker than those of <fr:link
-href="sub-000G.xml"
 type="local"
-addr="sub-000G">§I.6</fr:link>. However, it will be shown in the
+href="sub-000G.xml"
+addr="sub-000G"
+title="Topos à la Sheaves in Geometry and Logic: A First Introduction to Topos Theory p. 48">§I.6</fr:link>. However, it will be shown in the
             course of this chapter that the two sets of axioms are actually
             equivalent.
         </fr:p><fr:p>
             Notice that the two natural isomorphisms (1) and (2) may be combined
             in a single isomorphism
             <fr:tex
-display="block">                  \begin {align*}                     Sub_{ \mathcal {E} }(B × A) &amp;≅ Hom_{ \mathcal {E} }(A, PB) &amp;(3)  \\                   \end {align*}             </fr:tex>
-            natural in A. With <fr:tex>B = 1</fr:tex>, this implies (1) with <fr:tex>Ω = P1</fr:tex>, and hence
+display="block"><![CDATA[                 \begin {align*}                     Sub_{\mathcal {E}}(B × A) &≅ Hom_{\mathcal {E}}(A, PB) &(3) \\                 \end {align*}             ]]></fr:tex>
+            natural in A. With <fr:tex
+display="inline"><![CDATA[B = 1]]></fr:tex>, this implies (1) with <fr:tex
+display="inline"><![CDATA[Ω = P1]]></fr:tex>, and hence
             also (2). In other words, we could define a topos as a category with
-            finite limits equipped with a single operation <fr:tex>P</fr:tex> satisfying (3)."
-        </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:p>
+            finite limits equipped with a single operation <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> satisfying (3)."
+        </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     We collect the definition of a <fr:em>subobject classifier</fr:em>, which appears in
     every definition, and give a standard one:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>301</fr:anchor><fr:taxon>Definition</fr:taxon><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    For a category <fr:tex>\mathcal {C}</fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex>Ω</fr:tex> together
-    with a global element <fr:tex>true : 1 → Ω</fr:tex> such that <fr:tex>∀ A ∈  \mathcal {C} . ∀ s : S ↣ A. ∃!     χ_s : A → Ω.</fr:tex>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>251</fr:anchor><fr:addr>hejohns-000I</fr:addr><fr:route>hejohns-000I.xml</fr:route><fr:title>Subobject classifier</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+    For a category <fr:tex
+display="inline"><![CDATA[\mathcal {C}]]></fr:tex>, a <fr:em>subobject classifier</fr:em> is an object <fr:tex
+display="inline"><![CDATA[Ω]]></fr:tex> together
+    with a global element <fr:tex
+display="inline"><![CDATA[true : 1 → Ω]]></fr:tex> such that <fr:tex
+display="inline"><![CDATA[∀ A ∈ \mathcal {C}. ∀ s : S ↣ A. ∃!     χ_s : A → Ω.]]></fr:tex>
     <fr:embedded-tex
-hash="e02927923b00b58796778a2184dd7584"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-         \begin {tikzcd}
-            S  \ar [d, rightarrowtail, "s"'] \ar [r] \ar [dr, phantom, " \lrcorner ", very near start]  &amp; 1  \ar [d, rightarrowtail, "true"]  \\ 
-            A  \ar [r, "χ_s"'] &amp; Ω
-         \end {tikzcd}
-    </fr:embedded-tex-body></fr:embedded-tex>
+hash="1fb1dbf087f1accd0542d92729e7da0d"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+        \begin {tikzcd}
+            S \ar [d, rightarrowtail, "s"']\ar [r]\ar [dr, phantom, "\lrcorner ", very near start] & 1 \ar [d, rightarrowtail, "true"] \\
+            A \ar [r, "χ_s"'] & Ω
+        \end {tikzcd}
+    ]]></fr:embedded-tex-body></fr:embedded-tex>
     is a pullback square.
 </fr:p><fr:p>
-    Notice that <fr:tex>true</fr:tex> is trivially monic since every <fr:link
-href="hejohns-000K.xml"
+    Notice that <fr:tex
+display="inline"><![CDATA[true]]></fr:tex> is trivially monic since every <fr:link
 type="local"
-addr="hejohns-000K">global
+href="hejohns-000K.xml"
+addr="hejohns-000K"
+title="Global element">global
     element</fr:link> is trivially monic. Recall that "monos are stable
     under pullback".
 </fr:p><fr:p>
     This is from <fr:link
-href="mclarty-1992.xml"
 type="local"
-addr="mclarty-1992">Elementary Categories, Elementary Toposes</fr:link>.
-</fr:p></fr:mainmatter></fr:tree><fr:p>
+href="mclarty-1992.xml"
+addr="mclarty-1992"
+title="Elementary Categories, Elementary Toposes">Elementary Categories, Elementary Toposes</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:p>
     So comparing each definition with another, they're all about par for
     course, if we ignore natural numbers objects for now.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date></fr:frontmatter><fr:mainmatter></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/subb-0006.xml b/_site/forest/subb-0006.xml
index a1550f6..645ab81 100644
--- a/_site/forest/subb-0006.xml
+++ b/_site/forest/subb-0006.xml
@@ -1,218 +1,307 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>676</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>790</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>674</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>792</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>670</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>794</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
+        thus an adjunction core always determines an adjunction.
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
+        gives rise to a thin adjunction, as remarked <fr:link
+type="local"
+href="subb-0006.xml"
+addr="subb-0006">here</fr:link>.
+        I don't know how useful this lemma is for ordinary Homset categories.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>796</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>672</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>798</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     This is mentioned in passing, in the intro to <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">The Core of Adjoint Functors</fr:link>.
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">The Core of Adjoint Functors</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>P</fr:tex> and <fr:tex>Q</fr:tex> are posets, <fr:tex>f : P → Q</fr:tex> is a priori just a function, and <fr:tex>u : P ← Q</fr:tex> monotone,
-        such that <fr:tex>f(x) ≤ a ⟺ x ≤ u(a)</fr:tex>.
-        Then <fr:tex>f</fr:tex> monotone as well.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[P]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[Q]]></fr:tex> are posets, <fr:tex
+display="inline"><![CDATA[f : P → Q]]></fr:tex> is a priori just a function, and <fr:tex
+display="inline"><![CDATA[u : P ← Q]]></fr:tex> monotone,
+        such that <fr:tex
+display="inline"><![CDATA[f(x) ≤ a ⟺ x ≤ u(a)]]></fr:tex>.
+        Then <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone as well.
     </fr:p><fr:p>
-        Or dually, suppose that <fr:tex>f</fr:tex> monotone, but <fr:tex>u</fr:tex> is a priori just a function.
-        Then <fr:tex>u</fr:tex> monotone as well.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Or dually, suppose that <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> monotone, but <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is a priori just a function.
+        Then <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> monotone as well.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Suppose <fr:tex>p₁ ≤ p₂</fr:tex>.
-        Notice that <fr:tex>p₂ ≤ u∘f(p₂)</fr:tex>, by the unit of the (thin) adjunction.
-        Then <fr:tex>p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        Suppose <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[p₂ ≤ u∘f(p₂)]]></fr:tex>, by the unit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[p₁ ≤ p₂ ≤ u∘f(p₂) ⟹ f(p₁) ≤ f(p₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
     </fr:p><fr:p>
         (dual)
-        Suppose <fr:tex>q₁ ≤ q₂</fr:tex>.
-        Notice that <fr:tex>f∘u(q₁) ≤ q₂</fr:tex> by the counit of the (thin) adjunction.
-        Then <fr:tex>f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)</fr:tex> by the Hom bijection of the (thin) adjunction.
-    </fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+        Suppose <fr:tex
+display="inline"><![CDATA[q₁ ≤ q₂]]></fr:tex>.
+        Notice that <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₂]]></fr:tex> by the counit of the (thin) adjunction.
+        Then <fr:tex
+display="inline"><![CDATA[f∘u(q₁) ≤ q₁ ≤ q₂ ⟹ u(q₁) ≤ u(q₂)]]></fr:tex> by the Hom bijection of the (thin) adjunction.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0006</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Notice that <fr:link
-href="sub-0006.xml"
 type="local"
-addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex>u</fr:tex> is monotone.
+href="sub-0006.xml"
+addr="sub-0006">the above proof</fr:link> does <fr:em>not</fr:em> use the information that <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> is monotone.
         Thus <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the above lemma</fr:link> holds where <fr:tex>f</fr:tex> and <fr:tex>u</fr:tex> are just a priori functions.
-        The <fr:link
 href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the above lemma</fr:link> holds where <fr:tex
+display="inline"><![CDATA[f]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[u]]></fr:tex> are just a priori functions.
+        The <fr:link
 type="local"
-addr="hejohns-0006">above lemma</fr:link> is in some sense the more
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">above lemma</fr:link> is in some sense the more
         immediately natural question, but can be easily strengthened to
         the form of this corollary.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>668</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
-        thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
-        gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
-type="local"
-addr="subb-0006">here</fr:link>.
-        I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/subb-000D.xml b/_site/forest/subb-000D.xml
index a19e323..b5a1cb5 100644
--- a/_site/forest/subb-000D.xml
+++ b/_site/forest/subb-000D.xml
@@ -1,236 +1,316 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>630</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>800</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>628</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>802</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -238,224 +318,304 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>626</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>804</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -463,7 +623,8 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/subb-000E.xml b/_site/forest/subb-000E.xml
index ac8214e..3fa9cce 100644
--- a/_site/forest/subb-000E.xml
+++ b/_site/forest/subb-000E.xml
@@ -1,519 +1,675 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>597</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>844</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title><fr:link
+type="local"
+href="unstable-214.xml"
+addr="#214">Untitled</fr:link> › I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>595</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>846</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>593</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>848</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/subbb-000D.xml b/_site/forest/subbb-000D.xml
index 625067e..b44a004 100644
--- a/_site/forest/subbb-000D.xml
+++ b/_site/forest/subbb-000D.xml
@@ -1,19 +1,22 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>638</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>806</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -21,224 +24,304 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>636</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>808</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -246,282 +329,304 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>634</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>632</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>810</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -529,7 +634,87 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
+type="local"
 href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Related</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>812</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
+            <fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
+            <fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/subbb-000E.xml b/_site/forest/subbb-000E.xml
index e379ffa..8334f24 100644
--- a/_site/forest/subbb-000E.xml
+++ b/_site/forest/subbb-000E.xml
@@ -1,519 +1,675 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>603</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>850</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title><fr:link
+type="local"
+href="unstable-214.xml"
+addr="#214">Untitled</fr:link> › II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>601</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>852</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>599</fr:anchor><fr:taxon>Lemma</fr:taxon><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>854</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>341</fr:anchor><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 href="street-2012.xml"
-type="local"
-addr="street-2012">"An adjunction core consists of"</fr:link>
+addr="street-2012"
+title="The Core of Adjoint Functors">"An adjunction core consists of"</fr:link>
         <fr:ul><fr:li>
-                functions <fr:tex>U : A .ob → X .ob</fr:tex> and <fr:tex>F : X .ob → A .ob</fr:tex></fr:li>
+                functions <fr:tex
+display="inline"><![CDATA[U : A .ob → X .ob]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[F : X .ob → A .ob]]></fr:tex></fr:li>
             <fr:li>
-                a family of isomorphisms <fr:tex>(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}</fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+                a family of isomorphisms <fr:tex
+display="inline"><![CDATA[(π_{x,a} : A(F(x), a) ≅ X(X, U(a)))_{a ∈ A, x ∈ X}]]></fr:tex></fr:li></fr:ul></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        An adjunction core extends to an adjuction iff <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        An adjunction core extends to an adjuction iff <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex>
         <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> commutes, and iff <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. a ∈ A.]]></fr:tex> <fr:link
 type="local"
-addr="subb-000E">I</fr:link> commutes, and iff <fr:tex>∀x,y ∈ X. a ∈ A.</fr:tex> <fr:link
 href="subbb-000E.xml"
-type="local"
-addr="subbb-000E">II</fr:link>
+addr="subbb-000E"
+title="Untitled › II">II</fr:link>
         commutes.
         The adjunction is unique when it exists.
     </fr:p><fr:p>
         The <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">original theorem</fr:link> is stated in terms of
-        <fr:tex>\mathcal {V}</fr:tex>-enriched categories, but we only treat the simple case
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">original theorem</fr:link> is stated in terms of
+        <fr:tex
+display="inline"><![CDATA[\mathcal {V}]]></fr:tex>-enriched categories, but we only treat the simple case
         of Homset-categories here.
     </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="a09fe1256faafb1bc13a1cc059df3388"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F x ,  a )  \ar [r, " \overline {U} ×π_{ a , x }"] \ar [d, "∘"] &amp; X(U a , U b )×X( x , U a )  \ar [d, "∘"]  \\ 
-                A(F x ,  b )  \ar [r, "π_{ x ,  b }"] &amp; X( x , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="b806fb8c112337fe6c4b4a1e244b5d97"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(Fx, a) \ar [r, "\overline {U}×π_{a,x}"]\ar [d, "∘"] & X(Ua, Ub)×X(x, Ua) \ar [d, "∘"] \\
+                A(Fx, b) \ar [r, "π_{x, b}"] & X(x, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>#214</fr:parent></fr:frontmatter><fr:mainmatter>
         <fr:embedded-tex
-hash="8fd7996c4b1a3e39d4bda43697f06cf9"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                X( y , U a )×X( x ,  y )  \ar [r, "π_{ y ,  a }⁻¹× \overline {F} "] \ar [d, "∘"] &amp; A(F y ,  a )×A(F x , F y )  \ar [d, "∘"]  \\ 
-                X( x , U a )  \ar [r, "π_{x, a}⁻¹"] &amp; A(F x ,  a )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
-</fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+hash="802fd9b68c66249dd8fb86046d5d2e19"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                X(y, Ua)×X(x, y) \ar [r, "π_{y, a}⁻¹×\overline {F}"]\ar [d, "∘"] & A(Fy, a)×A(Fx, Fy) \ar [d, "∘"] \\
+                X(x, Ua) \ar [r, "π_{x, a}⁻¹"] & A(Fx, a)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
+</fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        We use the same notation and definitions of <fr:tex
+display="inline"><![CDATA[η, ϵ, \overline {F}, \overline {U}]]></fr:tex> as <fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        We use the same notation and definitions of <fr:tex>η, ϵ,  \overline {F} ,  \overline {U}</fr:tex> as <fr:link
 href="hejohns-000D.xml"
-type="local"
-addr="hejohns-000D">the
+addr="hejohns-000D"
+title="The verbose proof">the
         explicit proof of the precursor lemma</fr:link>.
-        The definition of <fr:tex>\overline {F} ,  \overline {U}</fr:tex> in <fr:link
-href="sub-0007.xml"
+        The definition of <fr:tex
+display="inline"><![CDATA[\overline {F}, \overline {U}]]></fr:tex> in <fr:link
 type="local"
-addr="sub-0007">the short proof</fr:link>
-        presupposes that <fr:tex>U</fr:tex> is a functor, so we're scrupulous to avoid
+href="sub-0007.xml"
+addr="sub-0007"
+title="Precursor to The Core of Adjoint Functors › the short proof (sketch)">the short proof</fr:link>
+        presupposes that <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> is a functor, so we're scrupulous to avoid
         circularity.
     </fr:p><fr:p>
         This proof is rather low tech and is lifted straight from <fr:link
-href="street-2012.xml"
 type="local"
-addr="street-2012">the
+href="street-2012.xml"
+addr="street-2012"
+title="The Core of Adjoint Functors">the
         paper</fr:link>. It's also how you'd attack <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
         precursor</fr:link> head-on. The utility is that it holds in
         enriched categories.
     </fr:p><fr:p>
         The forward directions are trivial.
-        Suppose <fr:tex>(F, U, (π_{a, x})_{a ∈ A, x ∈ X})</fr:tex> extends to <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+        Suppose <fr:tex
+display="inline"><![CDATA[(F, U, (π_{a, x})_{a ∈ A, x ∈ X})]]></fr:tex> extends to <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
         Then <fr:link
-href="subb-000E.xml"
 type="local"
-addr="subb-000E">I</fr:link> expresses that <fr:tex>∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) =  \overline {U} (g)∘π_{a, x}(f)</fr:tex>. This is exactly the condition
-        that <fr:tex>∀g : A(a, b). π</fr:tex> is natural in <fr:tex>a</fr:tex> at <fr:tex>g</fr:tex>, which is certainly
+href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). ∀f : A(F(x), a).         π_{x, b}(g∘f) = \overline {U}(g)∘π_{a, x}(f)]]></fr:tex>. This is exactly the condition
+        that <fr:tex
+display="inline"><![CDATA[∀g : A(a, b). π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> at <fr:tex
+display="inline"><![CDATA[g]]></fr:tex>, which is certainly
         true of an adjunction.
         <fr:link
-href="subbb-000E.xml"
 type="local"
-addr="subbb-000E">II</fr:link> expresses something similar (dual).
+href="subbb-000E.xml"
+addr="subbb-000E"
+title="Untitled › II">II</fr:link> expresses something similar (dual).
     </fr:p><fr:p>
         The nontrivial directions are the backward directions.
     </fr:p><fr:ul><fr:li><fr:p>
-                (I) wts I <fr:tex>⟹  \overline {U}</fr:tex> is a functor.
-                Suppose <fr:tex>∀a,b ∈ A. ∀x ∈ X.</fr:tex> I holds.
-                Then in particular, <fr:tex>∀c ∈ A. </fr:tex>I holds at <fr:tex>x = U(c)</fr:tex>.
+                (I) wts I <fr:tex
+display="inline"><![CDATA[⟹ \overline {U}]]></fr:tex> is a functor.
+                Suppose <fr:tex
+display="inline"><![CDATA[∀a,b ∈ A. ∀x ∈ X.]]></fr:tex> I holds.
+                Then in particular, <fr:tex
+display="inline"><![CDATA[∀c ∈ A. ]]></fr:tex>I holds at <fr:tex
+display="inline"><![CDATA[x = U(c)]]></fr:tex>.
             </fr:p>
             
         <fr:embedded-tex
-hash="ac9736f09827370a30fbadded4abb953"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-             \begin {tikzcd}
-                A( a ,  b )×A(F Uc ,  a )  \ar [r, " \overline {U} ×π_{ a , Uc }"] \ar [d, "∘"] &amp; X(U a , U b )×X( Uc , U a )  \ar [d, "∘"]  \\ 
-                A(F Uc ,  b )  \ar [r, "π_{ Uc ,  b }"] &amp; X( Uc , U b )
-             \end {tikzcd}
-        </fr:embedded-tex-body></fr:embedded-tex>
+hash="e29a45e5c3dfdeb85542e204370f1350"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+            \begin {tikzcd}
+                A(a, b)×A(FUc, a) \ar [r, "\overline {U}×π_{a,Uc}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                A(FUc, b) \ar [r, "π_{Uc, b}"] & X(Uc, Ub)
+            \end {tikzcd}
+        ]]></fr:embedded-tex-body></fr:embedded-tex>
 
             <fr:p>
                 As does this trivial square:
             </fr:p>
             <fr:embedded-tex
-hash="aaf8d81120cfc4781697983614442d7a"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, "id×(-∘ϵ_c)"] \ar [d, "∘"] &amp; A(a, b)×A(FUc, a)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, "(-∘ϵ_c)"] &amp; A(FUc, b)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="09f36af7f81c413359f103936dde5f36"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "id×(-∘ϵ_c)"]\ar [d, "∘"] & A(a, b)×A(FUc, a) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "(-∘ϵ_c)"] & A(FUc, b)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 which when pasted together, tells us magically:
             </fr:p>
             <fr:embedded-tex
-hash="2dec512dfc1acbbf48b6d09e3842ea04"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(a, b)×A(c, a)  \ar [r, " \overline {U} × \overline {U} "] \ar [d, "∘"] &amp; X(Ua, Ub)×X(Uc, Ua)  \ar [d, "∘"]  \\ 
-                    A(c, b)  \ar [r, " \overline {U} "] &amp; X(Uc, Ub)
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="4de6c4a2a2cb4b6f0547069c1ccdc586"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(a, b)×A(c, a) \ar [r, "\overline {U}×\overline {U}"]\ar [d, "∘"] & X(Ua, Ub)×X(Uc, Ua) \ar [d, "∘"] \\
+                    A(c, b) \ar [r, "\overline {U}"] & X(Uc, Ub)
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             <fr:p>
                 Combined with <fr:link
-href="hejohns-000D.xml"
 type="local"
-addr="hejohns-000D">the earlier proof</fr:link> that <fr:tex>\overline {U} (1ₐ)                 = 1_{ \overline {U} (a)}</fr:tex> (which just follows by definition),
-                we've shown that <fr:tex>\overline {U}</fr:tex> defines a functorial action on morphisms.
+href="hejohns-000D.xml"
+addr="hejohns-000D"
+title="The verbose proof">the earlier proof</fr:link> that <fr:tex
+display="inline"><![CDATA[\overline {U}(1ₐ)                 = 1_{\overline {U}(a)}]]></fr:tex> (which just follows by definition),
+                we've shown that <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> defines a functorial action on morphisms.
                 Actually, we saw earlier that <fr:link
+type="local"
 href="subb-000E.xml"
+addr="subb-000E"
+title="Untitled › I">I</fr:link> expresses that
+                <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, so we've met the preconditions of <fr:link
 type="local"
-addr="subb-000E">I</fr:link> expresses that
-                <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>, so we've met the preconditions of <fr:link
 href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">the
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
                 precursor lemma</fr:link>, thus there is a unique
-                extension <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}  ⊣_π  \overline {U}</fr:tex>.
+                extension <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣_π \overline {U}]]></fr:tex>.
             </fr:p>
             <fr:p>
                 There's two ways ways to see the uniqueness situation:
             </fr:p>
             <fr:ul><fr:li>
-                    (An intermediate notion) <fr:tex>\overline {U}</fr:tex> is the unique extension of
-                    <fr:tex>U</fr:tex> to a functor st <fr:tex>π</fr:tex> is natural in <fr:tex>a</fr:tex>.
+                    (An intermediate notion) <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> is the unique extension of
+                    <fr:tex
+display="inline"><![CDATA[U]]></fr:tex> to a functor st <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
                     We've essentially seen this already, if not literally.
-                    For any functor <fr:tex>U'</fr:tex>, naturality in <fr:tex>a</fr:tex> requires the following square to hold for all <fr:tex>g : A(a, b)</fr:tex>:
+                    For any functor <fr:tex
+display="inline"><![CDATA[U']]></fr:tex>, naturality in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex> requires the following square to hold for all <fr:tex
+display="inline"><![CDATA[g : A(a, b)]]></fr:tex>:
                     <fr:embedded-tex
-hash="7bced74c5568df77fd9237f12eab5655"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                         \begin {tikzcd}
-                            A(FU'a, a)  \ar [r, "g∘-"] &amp; A(FU'a, b)  \ar [d, "≀"', "π"]  \\ 
-                            X(U'a, U'a)  \ar [r, "U'g∘-"] \ar [u, "≀"', "π⁻¹"] &amp; X(U'a, U'b)
-                         \end {tikzcd}
-                    </fr:embedded-tex-body></fr:embedded-tex>
-                    Thus, <fr:tex>∀g. U'(g) =  \overline {U} (g)</fr:tex>.
-                    Then, with <fr:tex>\overline {U}</fr:tex> a functor and <fr:tex>π</fr:tex> natural in <fr:tex>a</fr:tex>, <fr:link
-href="hejohns-0007.xml"
+hash="2811be9da07c73dde4ea7d726bca767c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                        \begin {tikzcd}
+                            A(FU'a, a) \ar [r, "g∘-"] & A(FU'a, b) \ar [d, "≀"', "π"] \\
+                            X(U'a, U'a) \ar [r, "U'g∘-"]\ar [u, "≀"', "π⁻¹"] & X(U'a, U'b)
+                        \end {tikzcd}
+                    ]]></fr:embedded-tex-body></fr:embedded-tex>
+                    Thus, <fr:tex
+display="inline"><![CDATA[∀g. U'(g) = \overline {U}(g)]]></fr:tex>.
+                    Then, with <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> a functor and <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> natural in <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>, <fr:link
 type="local"
-addr="hejohns-0007">the
-                    precursor</fr:link> gives us a unique <fr:tex>\overline {F}</fr:tex> st <fr:tex>\overline {F}                      ⊣  \overline {U}</fr:tex> uniquely.
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the
+                    precursor</fr:link> gives us a unique <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st <fr:tex
+display="inline"><![CDATA[\overline {F}                     ⊣ \overline {U}]]></fr:tex> uniquely.
                 </fr:li>
                 <fr:li>
-                    (Going backwards) Since the definitions of <fr:tex>\overline {F}</fr:tex> and
-                    <fr:tex>\overline {U}</fr:tex> (in <fr:link
-href="hejohns-0007.xml"
+                    (Going backwards) Since the definitions of <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> and
+                    <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> (in <fr:link
 type="local"
-addr="hejohns-0007">the precursor</fr:link>) are not dependent
-                    on the other functor, the total adjunction <fr:tex>\overline {F}  ⊣  \overline {U}</fr:tex> is
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precursor</fr:link>) are not dependent
+                    on the other functor, the total adjunction <fr:tex
+display="inline"><![CDATA[\overline {F} ⊣ \overline {U}]]></fr:tex> is
                     the unique one extending the core by applying the
                     uniqueness result of <fr:link
-href="hejohns-0007.xml"
 type="local"
-addr="hejohns-0007">the precusor</fr:link> to each
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">the precusor</fr:link> to each
                     functor in turn, just knowing the other is a well-defined
                     functor.
                 </fr:li></fr:ul></fr:li>
         <fr:li><fr:p>
-                (II) wts II <fr:tex>⟹  \overline {F}</fr:tex> is a functor.
+                (II) wts II <fr:tex
+display="inline"><![CDATA[⟹ \overline {F}]]></fr:tex> is a functor.
                 This is dual.
-            </fr:p></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+            </fr:p></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:taxon>Proof corollary</fr:taxon><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        (For <fr:tex>V</fr:tex>-enriched categories, ) If <fr:tex>V</fr:tex> is thin, all diagrams commute,
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000E</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+        (For <fr:tex
+display="inline"><![CDATA[V]]></fr:tex>-enriched categories, ) If <fr:tex
+display="inline"><![CDATA[V]]></fr:tex> is thin, all diagrams commute,
         thus an adjunction core always determines an adjunction.
-    </fr:p><fr:p><fr:tex>Set</fr:tex> isn't thin, but <fr:tex>2</fr:tex> is, thus for posets, an adjunction core
+    </fr:p><fr:p><fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> isn't thin, but <fr:tex
+display="inline"><![CDATA[2]]></fr:tex> is, thus for posets, an adjunction core
         gives rise to a thin adjunction, as remarked <fr:link
-href="subb-0006.xml"
 type="local"
+href="subb-0006.xml"
 addr="subb-0006">here</fr:link>.
         I don't know how useful this lemma is for ordinary Homset categories.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/subsub-0007.xml b/_site/forest/subsub-0007.xml
index 9830bf6..4581fec 100644
--- a/_site/forest/subsub-0007.xml
+++ b/_site/forest/subsub-0007.xml
@@ -1,235 +1,387 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>682</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>814</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>816</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+type="local"
+href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
+            <fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>678</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
 href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
+        </fr:li>
+
+        <fr:li>
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
+            <fr:tree
+toc="true"
+numbered="true"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
+toc="true"
+numbered="false"
+show-heading="true"
+show-metadata="true"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>818</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+type="local"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     We take inspiration from <fr:link
-href="hejohns-0006.xml"
 type="local"
-addr="hejohns-0006">the thin case</fr:link>.
-    We wts the <fr:tex>F</fr:tex>-action on morphisms is fully determined already (∃!).
+href="hejohns-0006.xml"
+addr="hejohns-0006"
+title="The core of adjoint monotone functions">the thin case</fr:link>.
+    We wts the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action on morphisms is fully determined already (∃!).
     That is, we have two goals:
     <fr:ul><fr:li>
-            (∃) Show that there is a suitable <fr:tex>F</fr:tex>-action.
+            (∃) Show that there is a suitable <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
         </fr:li>
         <fr:li>
-            (!) That any other <fr:tex>F</fr:tex>-action is <fr:em>equal</fr:em> to it.
+            (!) That any other <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action is <fr:em>equal</fr:em> to it.
         </fr:li></fr:ul>
-    As usual, let <fr:tex>η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))</fr:tex> denote the
+    As usual, let <fr:tex
+display="inline"><![CDATA[η_x = πₓ(F(x))(1_{F(x)}) : X(x, U∘F(x))]]></fr:tex> denote the
     component of the unit of the (a posteriori) adjunction
-    and <fr:tex>ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})</fr:tex> denote the component of the counit.
+    and <fr:tex
+display="inline"><![CDATA[ϵₐ = π_{U(a)}(a)⁻¹(1_{U(a)})]]></fr:tex> denote the component of the counit.
 </fr:p><fr:ul><fr:li>
         (∃)
         <fr:p>
-            Now suppose <fr:tex>∃f : X(x, y)</fr:tex>.
-            Since <fr:tex>η_y : X(y, U∘F(y))</fr:tex>, we have <fr:tex>η_y∘f : X(x, U∘F(y))</fr:tex>.
-            By the Hom bijection, <fr:tex>πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))</fr:tex>.
-            This is our candidate for the <fr:tex>F</fr:tex>-action.
-            <fr:tex>\overline {F} (f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)</fr:tex>.
+            Now suppose <fr:tex
+display="inline"><![CDATA[∃f : X(x, y)]]></fr:tex>.
+            Since <fr:tex
+display="inline"><![CDATA[η_y : X(y, U∘F(y))]]></fr:tex>, we have <fr:tex
+display="inline"><![CDATA[η_y∘f : X(x, U∘F(y))]]></fr:tex>.
+            By the Hom bijection, <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹(η_y∘f) : A(F(x), F(y))]]></fr:tex>.
+            This is our candidate for the <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>-action.
+            <fr:tex
+display="inline"><![CDATA[\overline {F}(f) ≔ πₓ(F(y))⁻¹(η_y ∘ f)]]></fr:tex>.
         </fr:p>
         <fr:p>
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives rise to the following
-            square in <fr:tex>Set</fr:tex>:
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives rise to the following
+            square in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="7da984d3ff2027c69ebefb406d434b29"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), F(y)) &amp; A(F(y), F(y))  \ar [d, "≀"', "π_y(F(y))"]  \\ 
-                        X(x, U∘F(y))  \ar [u, "≀"', "πₓ(F(y))⁻¹"] &amp; X(y, U∘F(y))  \ar [l, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            And our candidate <fr:tex>\overline {F} (f)</fr:tex> is the one we know to exist by
-            chasing <fr:tex>1_{F(y)}</fr:tex> around the square.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="3bbd5cc482a2937801440e7dc20c0690"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), F(y)) & A(F(y), F(y)) \ar [d, "≀"', "π_y(F(y))"] \\
+                        X(x, U∘F(y)) \ar [u, "≀"', "πₓ(F(y))⁻¹"] & X(y, U∘F(y)) \ar [l, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            And our candidate <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> is the one we know to exist by
+            chasing <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> around the square.
         </fr:p>
         <fr:p>
-            We could directly try to show that <fr:tex>\overline {F}</fr:tex> defined as such is
+            We could directly try to show that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defined as such is
             indeed functorial--and I <fr:em>have</fr:em> tried--, but it's rather
             difficult without the clarifying perspective of the Yoneda
             lemma (no, seriously, although I'm sure one could do it).
         </fr:p>
         <fr:p>
             It's instructive to try directly, if one hasn't.
-            The problem is that <fr:tex>\overline {F} (f)</fr:tex> as defined is not obviously the
+            The problem is that <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as defined is not obviously the
             composition of morphisms in any diagram-- how do you reason
-            about the functoriality of <fr:tex>πₓ(F(y))⁻¹</fr:tex>?
+            about the functoriality of <fr:tex
+display="inline"><![CDATA[πₓ(F(y))⁻¹]]></fr:tex>?
             Instead, we want to invoke a representation theorem and work
-            with <fr:tex>\overline {F} (f)</fr:tex> as an explicit composition in some diagram.
+            with <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit composition in some diagram.
             The solution is to follow the Yoneda principle and
-            pass from reasoning in <fr:tex>Set</fr:tex> to <fr:tex>Setᴬ</fr:tex>.
+            pass from reasoning in <fr:tex
+display="inline"><![CDATA[Set]]></fr:tex> to <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>.
         </fr:p>
         <fr:p>
             Notice that the <fr:link
-href="subb-000D.xml"
 type="local"
+href="subb-000D.xml"
 addr="subb-000D">above square</fr:link> is
-            moreover <fr:em>natural</fr:em> in the variable <fr:tex>F(y)</fr:tex>.
-            That is, we've actually described the <fr:tex>F(y)</fr:tex> component of the
-            natural transformation <fr:tex>πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)</fr:tex>.
-            As a picture, each <fr:tex>f : X(x, y)</fr:tex> gives
-            rise to the following square in <fr:tex>Setᴬ</fr:tex>:
+            moreover <fr:em>natural</fr:em> in the variable <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex>.
+            That is, we've actually described the <fr:tex
+display="inline"><![CDATA[F(y)]]></fr:tex> component of the
+            natural transformation <fr:tex
+display="inline"><![CDATA[πₓ⁻¹∘(-∘f)∘π_y : A(F(x), -) ⇐ A(F(y),             -)]]></fr:tex>.
+            As a picture, each <fr:tex
+display="inline"><![CDATA[f : X(x, y)]]></fr:tex> gives
+            rise to the following square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="6fafc80d9b9e7f0489fcf2637e5c0717"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"]  \\ 
-                        X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree></fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="15c5a25c9843cce64b041e5e69e926cf"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"] \\
+                        X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree></fr:p>
         <fr:p>
-            We first make the trivial observation that <fr:tex>A(F(x), -)</fr:tex> and
-            <fr:tex>A(F(y), -)</fr:tex> are both Yoneda embeddings (we use the "version"
-            where <fr:tex>よ : Aᵒᵖ → Setᴬ</fr:tex>, ie the Yoneda embedding on <fr:tex>Aᵒᵖ</fr:tex>),
-            <fr:tex>よ(F(x))</fr:tex> and <fr:tex>よ(F(y))</fr:tex> respectively.
+            We first make the trivial observation that <fr:tex
+display="inline"><![CDATA[A(F(x), -)]]></fr:tex> and
+            <fr:tex
+display="inline"><![CDATA[A(F(y), -)]]></fr:tex> are both Yoneda embeddings (we use the "version"
+            where <fr:tex
+display="inline"><![CDATA[よ : Aᵒᵖ → Setᴬ]]></fr:tex>, ie the Yoneda embedding on <fr:tex
+display="inline"><![CDATA[Aᵒᵖ]]></fr:tex>),
+            <fr:tex
+display="inline"><![CDATA[よ(F(x))]]></fr:tex> and <fr:tex
+display="inline"><![CDATA[よ(F(y))]]></fr:tex> respectively.
         </fr:p>
         <fr:ul><fr:li><fr:p>
                     Then the Yoneda lemma tells us that by specifying an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex>, this fully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex>, this fully
                     determines (ie freely extends to) a natural
                     transformation <fr:tex
-display="block">よ( \overline {F} (f)) = (-∘ \overline {F} (f)) : よ(F(x)) ⇐                     よ(F(y))</fr:tex></fr:p></fr:li>
+display="block"><![CDATA[よ(\overline {F}(f)) = (-∘\overline {F}(f)) : よ(F(x)) ⇐                     よ(F(y))]]></fr:tex></fr:p></fr:li>
             <fr:li><fr:p>
                     Equivalently, the Yoneda principle tells us that an
-                    element <fr:tex>\overline {F} (f) : A(F(x), F(y))</fr:tex> fully failthfully
+                    element <fr:tex
+display="inline"><![CDATA[\overline {F}(f) : A(F(x), F(y))]]></fr:tex> fully failthfully
                     embeds as a natural transformation <fr:tex
-display="block">よ( \overline {F} (f)) =                     (-∘ \overline {F} (f)) : よ(F(x)) ⇐ よ(F(y))</fr:tex></fr:p></fr:li></fr:ul>
+display="block"><![CDATA[よ(\overline {F}(f)) =                     (-∘\overline {F}(f)) : よ(F(x)) ⇐ よ(F(y))]]></fr:tex></fr:p></fr:li></fr:ul>
         <fr:p>
             With this in mind, we notice that by definition,
-            <fr:tex>よ( \overline {F} (f))</fr:tex> is exactly the lid to <fr:link
-href="sub-000D.xml"
+            <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(f))]]></fr:tex> is exactly the lid to <fr:link
 type="local"
-addr="sub-000D">the above <fr:tex>Setᴬ</fr:tex>
+href="sub-000D.xml"
+addr="sub-000D">the above <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex>
             square</fr:link>.
             <fr:embedded-tex
-hash="af1879add0ed781bb43f5010d17e0754"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}                 A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
-            That is, both natural transformations map <fr:tex>1_{F(y)}</fr:tex> to the
-            same thing-- <fr:tex>\overline {F} {F}(f)</fr:tex>-- by definition, thus must be equal
+hash="6a7be1450e162f11307400423dc3e4e1"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}                 A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀", "πₓ⁻¹"'] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
+            That is, both natural transformations map <fr:tex
+display="inline"><![CDATA[1_{F(y)}]]></fr:tex> to the
+            same thing-- <fr:tex
+display="inline"><![CDATA[\overline {F}{F}(f)]]></fr:tex>-- by definition, thus must be equal
             by the Yoneda Lemma.
             (<fr:link
-href="subbb-000D.xml"
 type="local"
+href="subbb-000D.xml"
 addr="subbb-000D">In hindsight</fr:link>, this line of reasoning is quite
             circumlocutious and you might as well start by saying such a
-            <fr:tex>\overline {F}</fr:tex> would have to satisfy this square, but I think it's the
+            <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> would have to satisfy this square, but I think it's the
             more obvious starting point without presupposing it exists.)
-            Now that we've represented <fr:tex>\overline {F} (f)</fr:tex> as an explicit
+            Now that we've represented <fr:tex
+display="inline"><![CDATA[\overline {F}(f)]]></fr:tex> as an explicit
             composition, we have enough to tools consider
             functoriality.
         </fr:p>
         <fr:p>
-            Wts <fr:tex>\overline {F}</fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex>F</fr:tex>.
-            Indeed, <fr:tex>\overline {F} (1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}</fr:tex> by definition.
-            And suppose <fr:tex>∃g : X(y, z)</fr:tex>.
-            Then <fr:tex>\overline {F} (g∘f : X(x, z))</fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
+            Wts <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> actually defines a functorial action on morphisms for a functor extending <fr:tex
+display="inline"><![CDATA[F]]></fr:tex>.
+            Indeed, <fr:tex
+display="inline"><![CDATA[\overline {F}(1ₓ : X(x, x)) =  πₓ(F(x))⁻¹(ηₓ∘1ₓ) = πₓ(F(x))⁻¹(ηₓ) = 1_{F(x)}]]></fr:tex> by definition.
+            And suppose <fr:tex
+display="inline"><![CDATA[∃g : X(y, z)]]></fr:tex>.
+            Then <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f : X(x, z))]]></fr:tex> viewed as a natural transformation gives rise to the following superimposed squares:
             <fr:embedded-tex
-hash="0dd47869e77364c0688d537d44f8f49f"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                 \begin {tikzcd}
-                    A(F(x), -) &amp; A(F(y), -)  \ar [d, Rightarrow, "π_y"', shift right=2] \ar [l, Rightarrow, "-∘ \overline {F} (f)"'] &amp; A(F(z), -)  \ar [d, Rightarrow, "π_z"] \ar [l, Rightarrow, "-∘ \tilde {F}(g)"'] \ar [ll, Rightarrow, "-∘ \overline {F} (g∘f)"', bend right]  \\ 
-                    X(x, U(-))  \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"] \ar [u, Rightarrow, "π_y⁻¹"', shift right=2] &amp; X(z, U(-))  \ar [l, Rightarrow, "-∘g"] \ar [ll, Rightarrow, "-∘(g∘f)", bend left]
-                 \end {tikzcd}
-            </fr:embedded-tex-body></fr:embedded-tex>
+hash="a1aae49cf27956251484176f63510158"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                \begin {tikzcd}
+                    A(F(x), -) & A(F(y), -) \ar [d, Rightarrow, "π_y"', shift right=2]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] & A(F(z), -) \ar [d, Rightarrow, "π_z"]\ar [l, Rightarrow, "-∘\tilde {F}(g)"']\ar [ll, Rightarrow, "-∘\overline {F}(g∘f)"', bend right] \\
+                    X(x, U(-)) \ar [u, Rightarrow, "≀"', "πₓ⁻¹"] & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]\ar [u, Rightarrow, "π_y⁻¹"', shift right=2] & X(z, U(-)) \ar [l, Rightarrow, "-∘g"]\ar [ll, Rightarrow, "-∘(g∘f)", bend left]
+                \end {tikzcd}
+            ]]></fr:embedded-tex-body></fr:embedded-tex>
             A diagram chase verifies that the diagram indeed commutes, and
-            that <fr:tex>よ( \overline {F} (g∘f)) = よ( \overline {F} (f))∘よ( \overline {F} (g))</fr:tex>.
-            By the ("going home" direction of the) Yoneda principle, <fr:tex>\overline {F} (g∘f) =  \overline {F} (f)∘ \overline {F} (g)</fr:tex>.
-            Thus at last, <fr:tex>\overline {F}</fr:tex> defines a functorial action on <fr:tex>X</fr:tex>-morphisms.
+            that <fr:tex
+display="inline"><![CDATA[よ(\overline {F}(g∘f)) = よ(\overline {F}(f))∘よ(\overline {F}(g))]]></fr:tex>.
+            By the ("going home" direction of the) Yoneda principle, <fr:tex
+display="inline"><![CDATA[\overline {F}(g∘f) = \overline {F}(f)∘\overline {F}(g)]]></fr:tex>.
+            Thus at last, <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> defines a functorial action on <fr:tex
+display="inline"><![CDATA[X]]></fr:tex>-morphisms.
         </fr:p>
         <fr:p>
-            We should also check that <fr:tex>\overline {F}</fr:tex> is actually adjoint to <fr:tex>U</fr:tex>--
-            ie, that <fr:tex>(πₓ)_{x ∈ X}</fr:tex> is indeed <fr:em>natural in the variable
-            <fr:tex>x</fr:tex></fr:em> as well as <fr:tex>a</fr:tex>.
-            By great good fortune, by defining <fr:tex>\overline {F}</fr:tex> st
-            <fr:tex>∀x,y ∈ X. ∀f ∈ X(x, y).</fr:tex>
+            We should also check that <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> is actually adjoint to <fr:tex
+display="inline"><![CDATA[U]]></fr:tex>--
+            ie, that <fr:tex
+display="inline"><![CDATA[(πₓ)_{x ∈ X}]]></fr:tex> is indeed <fr:em>natural in the variable
+            <fr:tex
+display="inline"><![CDATA[x]]></fr:tex></fr:em> as well as <fr:tex
+display="inline"><![CDATA[a]]></fr:tex>.
+            By great good fortune, by defining <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> st
+            <fr:tex
+display="inline"><![CDATA[∀x,y ∈ X. ∀f ∈ X(x, y).]]></fr:tex>
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
             it is indeed the case.
         </fr:p></fr:li>
     <fr:li>
         (!)
         <fr:p>
-            To finish off the result, we wts for any other <fr:tex>F'</fr:tex> extending
-            <fr:tex>F</fr:tex> to morphisms st <fr:tex>F' ⊣ U</fr:tex>, witnessed by <fr:tex>π : Xᵒᵖ → Setᴬ</fr:tex>,
-            that <fr:tex>F' =  \overline {F}</fr:tex>.
-            This simply follows from the observation that <fr:tex>∀f.</fr:tex> naturality
-            of <fr:tex>π</fr:tex> is exactly <fr:link
-href="subbbb-000D"
-type="external">this square</fr:link>, and forces
-            <fr:tex>よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ( \overline {F} (f))</fr:tex></fr:p></fr:li></fr:ul><fr:p>
+            To finish off the result, we wts for any other <fr:tex
+display="inline"><![CDATA[F']]></fr:tex> extending
+            <fr:tex
+display="inline"><![CDATA[F]]></fr:tex> to morphisms st <fr:tex
+display="inline"><![CDATA[F' ⊣ U]]></fr:tex>, witnessed by <fr:tex
+display="inline"><![CDATA[π : Xᵒᵖ → Setᴬ]]></fr:tex>,
+            that <fr:tex
+display="inline"><![CDATA[F' = \overline {F}]]></fr:tex>.
+            This simply follows from the observation that <fr:tex
+display="inline"><![CDATA[∀f.]]></fr:tex> naturality
+            of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> is exactly <fr:link
+type="external"
+href="subbbb-000D">this square</fr:link>, and forces
+            <fr:tex
+display="inline"><![CDATA[よ(F'(f)) = πₓ⁻¹∘(-∘f)∘π_y = よ(\overline {F}(f))]]></fr:tex></fr:p></fr:li></fr:ul><fr:p>
     So this result is really a corollary of the Yoneda lemma.
     <fr:link
-href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject"
-type="external">The nlab page is here</fr:link>.
+type="external"
+href="https://ncatlab.org/nlab/show/adjoint+functor#AdjointFunctorFromObjectwiseRepresentingObject">The nlab page is here</fr:link>.
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:taxon>Moral</fr:taxon><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-000D</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Of course in hindsight, we could've (or even should've)
-        <fr:em>just</fr:em> defined  <fr:tex>\overline {F}</fr:tex> <fr:em>to be</fr:em>
-        <fr:tex>よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex> from the get-go, since it's a much
+        <fr:em>just</fr:em> defined  <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> <fr:em>to be</fr:em>
+        <fr:tex
+display="inline"><![CDATA[よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex> from the get-go, since it's a much
         more clearly well-behaved definition, but unless you're
         already sold on the Yoneda principle, I find it a difficult
         leap without struggling with lower-level objects a bit
@@ -237,65 +389,8 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0
         I'm still in training.
     </fr:p><fr:p>
         This is the perspective taken in <fr:link
-href="sub-0007.xml"
 type="local"
-addr="sub-0007">the short proof</fr:link>.
-    </fr:p></fr:mainmatter></fr:tree></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>680</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
-href="hejohns-0007.xml"
-type="local"
-addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
-        </fr:li>
-
-        <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
-            <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
-show-metadata="false"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="sub-0007.xml"
+addr="sub-0007"
+title="The short proof (sketch)">the short proof</fr:link>.
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/subsubb-0007.xml b/_site/forest/subsubb-0007.xml
index 1a275f5..ad65f05 100644
--- a/_site/forest/subsubb-0007.xml
+++ b/_site/forest/subsubb-0007.xml
@@ -1,77 +1,99 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>686</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>820</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Context</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>684</fr:anchor><fr:taxon>Proof</fr:taxon><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>822</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title><fr:link
+type="local"
 href="hejohns-0007.xml"
+addr="hejohns-0007"
+title="Precursor to The Core of Adjoint Functors">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns-0007">Precursor to The Core of Adjoint Functors</fr:link> › the short proof (sketch)</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
-            Suppose such a <fr:tex>\overline {F}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀f : X(x, y).</fr:tex> this square in <fr:tex>Setᴬ</fr:tex> commutes:
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>hejohns-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {F}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀f : X(x, y).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Setᴬ]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="04c41799d5084f092a5df1e435c7b32e"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}                     A(F(x), -)  \ar [d, Rightarrow, "≀", "πₓ"'] &amp; A(F(y), -)  \ar [d, Rightarrow, "≀"', "π_y"] \ar [l, Rightarrow, "-∘ \overline {F} (f)"']  \\ 
-                        X(x, U(-)) &amp; X(y, U(-))  \ar [l, Rightarrow, "-∘f"]
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            Then clearly <fr:tex>∃! \overline {F} .∀f : X(x, y).  \overline {F} (f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)</fr:tex>.
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="72916137e0dd482f96165eaf52c81f4c"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}                     A(F(x), -) \ar [d, Rightarrow, "≀", "πₓ"'] & A(F(y), -) \ar [d, Rightarrow, "≀"', "π_y"]\ar [l, Rightarrow, "-∘\overline {F}(f)"'] \\
+                        X(x, U(-)) & X(y, U(-)) \ar [l, Rightarrow, "-∘f"]
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            Then clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {F}.∀f : X(x, y). \overline {F}(f) = よ⁻¹(πₓ⁻¹∘(-∘f)∘π_y)]]></fr:tex>.
         </fr:li>
 
         <fr:li>
-            (dual) Suppose such a <fr:tex>\overline {U}</fr:tex> exists.
-            Then naturality of <fr:tex>π</fr:tex> requires that
-            <fr:tex>∀u : A(a, b).</fr:tex> this square in <fr:tex>Set^{Xᵒᵖ}</fr:tex> commutes:
+            (dual) Suppose such a <fr:tex
+display="inline"><![CDATA[\overline {U}]]></fr:tex> exists.
+            Then naturality of <fr:tex
+display="inline"><![CDATA[π]]></fr:tex> requires that
+            <fr:tex
+display="inline"><![CDATA[∀u : A(a, b).]]></fr:tex> this square in <fr:tex
+display="inline"><![CDATA[Set^{Xᵒᵖ}]]></fr:tex> commutes:
             <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
-hash="8efe46a4066df01c4e80f07f43789adb"><fr:embedded-tex-preamble>\usepackage {tikz-cd} \usepackage {unicode-math}</fr:embedded-tex-preamble><fr:embedded-tex-body>
-                     \begin {tikzcd}
-                        A(F(-), a)  \ar [r, Rightarrow, "(u∘-)"] \ar [d, Rightarrow, "≀"', "πₐ"] &amp; A(F(-), b)  \ar [d, Rightarrow, "≀"', "π_b"]  \\ 
-                        X(-, U(a))  \ar [r, Rightarrow, "( \overline {U} (u)∘-)"] &amp; X(-, U(b))
-                     \end {tikzcd}
-                </fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter></fr:tree>
-            and clearly <fr:tex>∃! \overline {U} .∀u : A(a, b).  \overline {U} (u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)</fr:tex>.
-        </fr:li></fr:ul></fr:mainmatter></fr:tree></fr:context><fr:related></fr:related><fr:backlinks></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>sub-0007</fr:parent></fr:frontmatter><fr:mainmatter><fr:embedded-tex
+hash="d297a9e21f314451cdc6a305c57a3560"><fr:embedded-tex-preamble><![CDATA[\usepackage {tikz-cd}\usepackage {unicode-math}]]></fr:embedded-tex-preamble><fr:embedded-tex-body><![CDATA[
+                    \begin {tikzcd}
+                        A(F(-), a) \ar [r, Rightarrow, "(u∘-)"]\ar [d, Rightarrow, "≀"', "πₐ"] & A(F(-), b) \ar [d, Rightarrow, "≀"', "π_b"] \\
+                        X(-, U(a)) \ar [r, Rightarrow, "(\overline {U}(u)∘-)"] & X(-, U(b))
+                    \end {tikzcd}
+                ]]></fr:embedded-tex-body></fr:embedded-tex></fr:mainmatter><fr:backmatter /></fr:tree>
+            and clearly <fr:tex
+display="inline"><![CDATA[∃!\overline {U}.∀u : A(a, b). \overline {U}(u) = よ⁻¹(π_b∘(u∘-)∘πₐ⁻¹)]]></fr:tex>.
+        </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/unstable-163.xml b/_site/forest/unstable-163.xml
index 87ae133..6519529 100644
--- a/_site/forest/unstable-163.xml
+++ b/_site/forest/unstable-163.xml
@@ -39,7 +39,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
diff --git a/_site/forest/unstable-164.xml b/_site/forest/unstable-164.xml
index 9931b87..bbe840a 100644
--- a/_site/forest/unstable-164.xml
+++ b/_site/forest/unstable-164.xml
@@ -42,7 +42,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -63,7 +63,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
diff --git a/_site/forest/unstable-165.xml b/_site/forest/unstable-165.xml
index 2548e5a..a1109d6 100644
--- a/_site/forest/unstable-165.xml
+++ b/_site/forest/unstable-165.xml
@@ -53,7 +53,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -74,7 +74,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
diff --git a/_site/forest/unstable-166.xml b/_site/forest/unstable-166.xml
index 45e7b58..621345b 100644
--- a/_site/forest/unstable-166.xml
+++ b/_site/forest/unstable-166.xml
@@ -57,7 +57,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
diff --git a/_site/forest/unstable-167.xml b/_site/forest/unstable-167.xml
index da66dbf..8f90638 100644
--- a/_site/forest/unstable-167.xml
+++ b/_site/forest/unstable-167.xml
@@ -89,7 +89,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-168.xml b/_site/forest/unstable-168.xml
index 93d0f59..d5b5c51 100644
--- a/_site/forest/unstable-168.xml
+++ b/_site/forest/unstable-168.xml
@@ -44,7 +44,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -67,7 +67,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-169.xml b/_site/forest/unstable-169.xml
index c0ba96b..c7a2a87 100644
--- a/_site/forest/unstable-169.xml
+++ b/_site/forest/unstable-169.xml
@@ -90,7 +90,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -113,7 +113,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-170.xml b/_site/forest/unstable-170.xml
index d7b4ecb..234e705 100644
--- a/_site/forest/unstable-170.xml
+++ b/_site/forest/unstable-170.xml
@@ -39,7 +39,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
diff --git a/_site/forest/unstable-171.xml b/_site/forest/unstable-171.xml
index 4c00ddb..eac0bf5 100644
--- a/_site/forest/unstable-171.xml
+++ b/_site/forest/unstable-171.xml
@@ -44,7 +44,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>443</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
diff --git a/_site/forest/unstable-172.xml b/_site/forest/unstable-172.xml
index 2b650a6..856980c 100644
--- a/_site/forest/unstable-172.xml
+++ b/_site/forest/unstable-172.xml
@@ -72,7 +72,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
diff --git a/_site/forest/unstable-173.xml b/_site/forest/unstable-173.xml
index 21aaaab..6d48abb 100644
--- a/_site/forest/unstable-173.xml
+++ b/_site/forest/unstable-173.xml
@@ -7,18 +7,18 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>483</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title><fr:link
 type="local"
 href="index.xml"
 addr="index"
-title="Johnson He">Johnson He</fr:link> › Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+title="Johnson He">Johnson He</fr:link> › Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>223</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1173,7 +1173,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>270</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1184,7 +1184,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1201,7 +1201,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1228,7 +1228,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1254,7 +1254,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1285,7 +1285,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1302,7 +1302,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1346,7 +1346,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1365,7 +1365,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1394,7 +1394,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1414,7 +1414,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1467,7 +1467,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1527,7 +1527,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1642,7 +1642,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1684,7 +1684,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1721,7 +1721,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1732,7 +1732,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1754,7 +1754,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1789,7 +1789,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1808,7 +1808,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1827,7 +1827,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2015,7 +2015,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2046,7 +2046,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="true"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>484</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:meta
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:meta
 name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
@@ -2056,7 +2056,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>219</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2098,14 +2098,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>223</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3260,7 +3260,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>270</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3271,7 +3271,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3288,7 +3288,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3315,7 +3315,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3341,7 +3341,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3372,7 +3372,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3389,7 +3389,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3433,7 +3433,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3452,7 +3452,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3481,7 +3481,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3501,7 +3501,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3554,7 +3554,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3614,7 +3614,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3729,7 +3729,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3771,7 +3771,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3808,7 +3808,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3819,7 +3819,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3841,7 +3841,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3876,7 +3876,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3895,7 +3895,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3914,7 +3914,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4102,7 +4102,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4128,21 +4128,21 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
     course text(s). I haven't used it much for reading either, owing to the
     large amount of new notation one has to refresh every time you open the
@@ -4154,14 +4154,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -4170,14 +4170,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>408</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -4187,14 +4187,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -4249,28 +4249,28 @@ show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
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 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
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 show-metadata="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -4290,7 +4290,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -4306,7 +4306,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Bibliographical Note: This Dover edition, first published in 2006, is a
     slightly corrected unabridged republication of the revised (second) edition
     of the work originally published in 1984 by Elsevier Science Publishers
@@ -4320,7 +4320,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>416</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -4342,21 +4342,21 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -4392,14 +4392,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -4437,14 +4437,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>422</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -4472,7 +4472,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>424</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -4501,14 +4501,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -4531,7 +4531,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4596,14 +4596,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -4624,7 +4624,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -4656,14 +4656,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>432</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -4682,14 +4682,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>434</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>221</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -4702,7 +4702,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -4718,7 +4718,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 type="local"
@@ -4738,7 +4738,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>436</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -4747,7 +4747,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
     this list. Whatever else you might think about Quine, he was a terrific
     explainer, and this is the best introduction to the subject I know."
@@ -4761,7 +4761,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4829,7 +4829,7 @@ show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>440</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 toc="true"
@@ -4838,7 +4838,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -4854,14 +4854,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>442</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>443</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -4919,7 +4919,7 @@ show-heading="true"
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 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4930,7 +4930,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4950,7 +4950,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>448</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4962,15 +4962,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>450</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        TODO: not a good selection
-    </fr:p><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Contributing to <fr:link
 type="external"
-href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+href="https://github.com/maxsnew/cubical-categorical-logic">Cubical Categorical Logic</fr:link>, "cubical agda formalizations of different approaches to categorical semantics of type theories"
         </fr:li>
         <fr:li><fr:link
 type="external"
-href="https://github.com/hejohns/.rc/tree/master/home/bin">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly
         </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
@@ -4978,7 +4977,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>451</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 type="external"
 href="https://mplse-reading-group.github.io/">https://mplse-reading-group.github.io/</fr:link></fr:li>
         <fr:li><fr:link
@@ -4990,7 +4989,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>452</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
 type="external"
 href="../posts.html">Jekyll Blog</fr:link></fr:mainmatter><fr:backmatter /></fr:tree>
     
@@ -5001,14 +5000,14 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>453</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5019,7 +5018,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>456</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
 type="external"
@@ -5030,7 +5029,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5049,7 +5048,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5063,7 +5062,7 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5093,7 +5092,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5109,7 +5108,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5124,7 +5123,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -6279,7 +6278,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>461</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -6289,14 +6288,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>462</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -6351,7 +6350,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>463</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -6367,7 +6366,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>464</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -6389,14 +6388,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>465</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -6432,14 +6431,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>466</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -6477,7 +6476,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>467</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -6490,14 +6489,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>469</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -6520,7 +6519,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -6585,7 +6584,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -6617,14 +6616,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>472</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -6645,7 +6644,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -6677,14 +6676,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>473</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -6703,7 +6702,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-174.xml b/_site/forest/unstable-174.xml
index 0b79d62..60b7d24 100644
--- a/_site/forest/unstable-174.xml
+++ b/_site/forest/unstable-174.xml
@@ -7,25 +7,25 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>408</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title><fr:link
 type="local"
 href="index.xml"
 addr="index"
-title="Johnson He">Johnson He</fr:link> › References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+title="Johnson He">Johnson He</fr:link> › References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
     course text(s). I haven't used it much for reading either, owing to the
     large amount of new notation one has to refresh every time you open the
@@ -37,14 +37,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -53,14 +53,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>408</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -70,14 +70,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -132,28 +132,28 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -173,7 +173,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -189,7 +189,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Bibliographical Note: This Dover edition, first published in 2006, is a
     slightly corrected unabridged republication of the revised (second) edition
     of the work originally published in 1984 by Elsevier Science Publishers
@@ -203,7 +203,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>416</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -225,21 +225,21 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -275,14 +275,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -320,14 +320,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>422</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -355,7 +355,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>424</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -384,14 +384,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -414,7 +414,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -479,14 +479,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -507,7 +507,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -539,14 +539,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>432</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -565,14 +565,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>434</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>221</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -585,7 +585,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -601,7 +601,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 type="local"
@@ -621,7 +621,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>436</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -630,7 +630,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
     this list. Whatever else you might think about Quine, he was a terrific
     explainer, and this is the best introduction to the subject I know."
@@ -644,7 +644,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -712,7 +712,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>440</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 toc="true"
@@ -721,7 +721,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -737,14 +737,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>442</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>443</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -802,7 +802,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -813,7 +813,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -833,7 +833,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>448</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -851,7 +851,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="true"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>486</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:meta
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:meta
 name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
@@ -861,7 +861,7 @@ show-heading="true"
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 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>219</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -903,14 +903,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>223</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2065,7 +2065,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>270</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2076,7 +2076,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
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 addr="hejohns"
@@ -2093,7 +2093,7 @@ show-heading="true"
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 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2120,7 +2120,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2146,7 +2146,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2177,7 +2177,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2194,7 +2194,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2238,7 +2238,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2257,7 +2257,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2286,7 +2286,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2306,7 +2306,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2359,7 +2359,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2419,7 +2419,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2534,7 +2534,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2576,7 +2576,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2613,7 +2613,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2624,7 +2624,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2646,7 +2646,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2681,7 +2681,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2700,7 +2700,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2719,7 +2719,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2907,7 +2907,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2933,21 +2933,21 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
     course text(s). I haven't used it much for reading either, owing to the
     large amount of new notation one has to refresh every time you open the
@@ -2959,14 +2959,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
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 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -2975,14 +2975,14 @@ show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>408</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -2992,14 +2992,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -3054,28 +3054,28 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -3095,7 +3095,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -3111,7 +3111,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Bibliographical Note: This Dover edition, first published in 2006, is a
     slightly corrected unabridged republication of the revised (second) edition
     of the work originally published in 1984 by Elsevier Science Publishers
@@ -3125,7 +3125,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>416</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -3147,21 +3147,21 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -3197,14 +3197,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -3242,14 +3242,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>422</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -3277,7 +3277,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>424</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -3306,14 +3306,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -3336,7 +3336,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3401,14 +3401,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -3429,7 +3429,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -3461,14 +3461,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>432</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -3487,14 +3487,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>434</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>221</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -3507,7 +3507,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -3523,7 +3523,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 type="local"
@@ -3543,7 +3543,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>436</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -3552,7 +3552,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
     this list. Whatever else you might think about Quine, he was a terrific
     explainer, and this is the best introduction to the subject I know."
@@ -3566,7 +3566,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3634,7 +3634,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>440</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 toc="true"
@@ -3643,7 +3643,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -3659,14 +3659,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>442</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>443</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -3724,7 +3724,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3735,7 +3735,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3755,7 +3755,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>448</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3767,15 +3767,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>450</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        TODO: not a good selection
-    </fr:p><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Contributing to <fr:link
 type="external"
-href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+href="https://github.com/maxsnew/cubical-categorical-logic">Cubical Categorical Logic</fr:link>, "cubical agda formalizations of different approaches to categorical semantics of type theories"
         </fr:li>
         <fr:li><fr:link
 type="external"
-href="https://github.com/hejohns/.rc/tree/master/home/bin">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly
         </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
@@ -3783,7 +3782,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>451</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 type="external"
 href="https://mplse-reading-group.github.io/">https://mplse-reading-group.github.io/</fr:link></fr:li>
         <fr:li><fr:link
@@ -3795,7 +3794,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>452</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
 type="external"
 href="../posts.html">Jekyll Blog</fr:link></fr:mainmatter><fr:backmatter /></fr:tree>
     
@@ -3806,14 +3805,14 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>453</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3824,7 +3823,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>456</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
 type="external"
@@ -3835,7 +3834,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3854,7 +3853,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3868,7 +3867,7 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3898,7 +3897,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3914,7 +3913,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3929,7 +3928,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5084,7 +5083,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>461</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -5094,14 +5093,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>462</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -5156,7 +5155,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>463</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -5172,7 +5171,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>464</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -5194,14 +5193,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>465</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -5237,14 +5236,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>466</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -5282,7 +5281,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>467</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5295,14 +5294,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>469</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -5325,7 +5324,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5390,7 +5389,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5422,14 +5421,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>472</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -5450,7 +5449,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -5482,14 +5481,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>473</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -5508,7 +5507,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-175.xml b/_site/forest/unstable-175.xml
index ae1a753..1b4219b 100644
--- a/_site/forest/unstable-175.xml
+++ b/_site/forest/unstable-175.xml
@@ -7,19 +7,18 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>476</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title><fr:link
 type="local"
 href="index.xml"
 addr="index"
-title="Johnson He">Johnson He</fr:link> › Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        TODO: not a good selection
-    </fr:p><fr:ul><fr:li><fr:link
+title="Johnson He">Johnson He</fr:link> › Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Contributing to <fr:link
 type="external"
-href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+href="https://github.com/maxsnew/cubical-categorical-logic">Cubical Categorical Logic</fr:link>, "cubical agda formalizations of different approaches to categorical semantics of type theories"
         </fr:li>
         <fr:li><fr:link
 type="external"
-href="https://github.com/hejohns/.rc/tree/master/home/bin">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly
         </fr:li></fr:ul></fr:mainmatter><fr:backmatter><fr:tree
 toc="false"
 numbered="false"
@@ -34,7 +33,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="true"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>477</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:meta
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>400</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:meta
 name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
@@ -44,7 +43,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>219</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -86,14 +85,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>223</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1248,7 +1247,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>270</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1259,7 +1258,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1276,7 +1275,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1303,7 +1302,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1329,7 +1328,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1360,7 +1359,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1377,7 +1376,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1421,7 +1420,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1440,7 +1439,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1469,7 +1468,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1489,7 +1488,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1542,7 +1541,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1602,7 +1601,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1717,7 +1716,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1759,7 +1758,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1796,7 +1795,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1807,7 +1806,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1829,7 +1828,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1864,7 +1863,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1883,7 +1882,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1902,7 +1901,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2090,7 +2089,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2116,21 +2115,21 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
     course text(s). I haven't used it much for reading either, owing to the
     large amount of new notation one has to refresh every time you open the
@@ -2142,14 +2141,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
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 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -2158,14 +2157,14 @@ show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>408</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -2175,14 +2174,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -2237,28 +2236,28 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -2278,7 +2277,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -2294,7 +2293,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Bibliographical Note: This Dover edition, first published in 2006, is a
     slightly corrected unabridged republication of the revised (second) edition
     of the work originally published in 1984 by Elsevier Science Publishers
@@ -2308,7 +2307,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>416</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -2330,21 +2329,21 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -2380,14 +2379,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -2425,14 +2424,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>422</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -2460,7 +2459,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>424</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -2489,14 +2488,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -2519,7 +2518,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2584,14 +2583,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -2612,7 +2611,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -2644,14 +2643,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>432</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -2670,14 +2669,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>434</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>221</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -2690,7 +2689,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -2706,7 +2705,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 type="local"
@@ -2726,7 +2725,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>436</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -2735,7 +2734,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
     this list. Whatever else you might think about Quine, he was a terrific
     explainer, and this is the best introduction to the subject I know."
@@ -2749,7 +2748,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2817,7 +2816,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>440</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 toc="true"
@@ -2826,7 +2825,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -2842,14 +2841,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>442</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>443</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -2907,7 +2906,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2918,7 +2917,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2938,7 +2937,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>448</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2950,15 +2949,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>450</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        TODO: not a good selection
-    </fr:p><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Contributing to <fr:link
 type="external"
-href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+href="https://github.com/maxsnew/cubical-categorical-logic">Cubical Categorical Logic</fr:link>, "cubical agda formalizations of different approaches to categorical semantics of type theories"
         </fr:li>
         <fr:li><fr:link
 type="external"
-href="https://github.com/hejohns/.rc/tree/master/home/bin">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly
         </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
@@ -2966,7 +2964,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>451</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 type="external"
 href="https://mplse-reading-group.github.io/">https://mplse-reading-group.github.io/</fr:link></fr:li>
         <fr:li><fr:link
@@ -2978,7 +2976,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>452</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
 type="external"
 href="../posts.html">Jekyll Blog</fr:link></fr:mainmatter><fr:backmatter /></fr:tree>
     
@@ -2989,14 +2987,14 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>453</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3007,7 +3005,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>456</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
 type="external"
@@ -3018,7 +3016,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3037,7 +3035,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3051,7 +3049,7 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3081,7 +3079,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3097,7 +3095,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3112,7 +3110,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4267,7 +4265,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>461</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -4277,14 +4275,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>462</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -4339,7 +4337,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>463</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -4355,7 +4353,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>464</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -4377,14 +4375,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>465</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -4420,14 +4418,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>466</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -4465,7 +4463,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>467</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4478,14 +4476,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>469</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -4508,7 +4506,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4573,7 +4571,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4605,14 +4603,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>472</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -4633,7 +4631,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -4665,14 +4663,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>473</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -4691,7 +4689,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-176.xml b/_site/forest/unstable-176.xml
index 0d64867..7577e06 100644
--- a/_site/forest/unstable-176.xml
+++ b/_site/forest/unstable-176.xml
@@ -7,11 +7,11 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>481</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title><fr:link
 type="local"
 href="index.xml"
 addr="index"
-title="Johnson He">Johnson He</fr:link> › Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+title="Johnson He">Johnson He</fr:link> › Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 type="external"
 href="https://mplse-reading-group.github.io/">https://mplse-reading-group.github.io/</fr:link></fr:li>
         <fr:li><fr:link
@@ -30,7 +30,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="true"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>482</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:meta
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:meta
 name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
@@ -40,7 +40,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>219</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -82,14 +82,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>223</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1244,7 +1244,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>270</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1255,7 +1255,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1272,7 +1272,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1299,7 +1299,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1325,7 +1325,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1356,7 +1356,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1373,7 +1373,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1417,7 +1417,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1436,7 +1436,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1465,7 +1465,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1485,7 +1485,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1538,7 +1538,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1598,7 +1598,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1713,7 +1713,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1755,7 +1755,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1792,7 +1792,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1803,7 +1803,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1825,7 +1825,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1860,7 +1860,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1879,7 +1879,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1898,7 +1898,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2086,7 +2086,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2112,21 +2112,21 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
     course text(s). I haven't used it much for reading either, owing to the
     large amount of new notation one has to refresh every time you open the
@@ -2138,14 +2138,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
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 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -2154,14 +2154,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>408</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -2171,14 +2171,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -2233,28 +2233,28 @@ show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -2274,7 +2274,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -2290,7 +2290,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Bibliographical Note: This Dover edition, first published in 2006, is a
     slightly corrected unabridged republication of the revised (second) edition
     of the work originally published in 1984 by Elsevier Science Publishers
@@ -2304,7 +2304,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>416</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -2326,21 +2326,21 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -2376,14 +2376,14 @@ show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -2421,14 +2421,14 @@ show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>422</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -2456,7 +2456,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>424</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -2485,14 +2485,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -2515,7 +2515,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2580,14 +2580,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -2608,7 +2608,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -2640,14 +2640,14 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>432</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -2666,14 +2666,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>434</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>221</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -2686,7 +2686,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -2702,7 +2702,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 type="local"
@@ -2722,7 +2722,7 @@ show-heading="true"
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 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>436</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -2731,7 +2731,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
     this list. Whatever else you might think about Quine, he was a terrific
     explainer, and this is the best introduction to the subject I know."
@@ -2745,7 +2745,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2813,7 +2813,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>440</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 toc="true"
@@ -2822,7 +2822,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -2838,14 +2838,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>442</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>443</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -2903,7 +2903,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2914,7 +2914,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2934,7 +2934,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>448</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2946,15 +2946,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>450</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        TODO: not a good selection
-    </fr:p><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Contributing to <fr:link
 type="external"
-href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+href="https://github.com/maxsnew/cubical-categorical-logic">Cubical Categorical Logic</fr:link>, "cubical agda formalizations of different approaches to categorical semantics of type theories"
         </fr:li>
         <fr:li><fr:link
 type="external"
-href="https://github.com/hejohns/.rc/tree/master/home/bin">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly
         </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
@@ -2962,7 +2961,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>451</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 type="external"
 href="https://mplse-reading-group.github.io/">https://mplse-reading-group.github.io/</fr:link></fr:li>
         <fr:li><fr:link
@@ -2974,7 +2973,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>452</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
 type="external"
 href="../posts.html">Jekyll Blog</fr:link></fr:mainmatter><fr:backmatter /></fr:tree>
     
@@ -2985,14 +2984,14 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>453</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3003,7 +3002,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>456</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
 type="external"
@@ -3014,7 +3013,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3033,7 +3032,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3047,7 +3046,7 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3077,7 +3076,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3093,7 +3092,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3108,7 +3107,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4263,7 +4262,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>461</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -4273,14 +4272,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>462</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -4335,7 +4334,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>463</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -4351,7 +4350,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>464</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -4373,14 +4372,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>465</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -4416,14 +4415,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>466</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -4461,7 +4460,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>467</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4474,14 +4473,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>469</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -4504,7 +4503,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4569,7 +4568,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4601,14 +4600,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>472</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -4629,7 +4628,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -4661,14 +4660,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>473</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -4687,7 +4686,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-177.xml b/_site/forest/unstable-177.xml
index 93afdde..07fcdba 100644
--- a/_site/forest/unstable-177.xml
+++ b/_site/forest/unstable-177.xml
@@ -7,11 +7,11 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>217</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title><fr:link
 type="local"
 href="index.xml"
 addr="index"
-title="Johnson He">Johnson He</fr:link> › Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
+title="Johnson He">Johnson He</fr:link> › Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
 type="external"
 href="../posts.html">Jekyll Blog</fr:link></fr:mainmatter><fr:backmatter><fr:tree
 toc="false"
@@ -27,7 +27,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="true"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:meta
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>218</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:meta
 name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
@@ -37,7 +37,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>219</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -79,14 +79,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>223</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1241,7 +1241,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>270</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1252,7 +1252,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1269,7 +1269,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1296,7 +1296,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1322,7 +1322,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1353,7 +1353,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1370,7 +1370,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1414,7 +1414,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1433,7 +1433,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1462,7 +1462,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1482,7 +1482,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1535,7 +1535,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1595,7 +1595,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1710,7 +1710,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1752,7 +1752,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1789,7 +1789,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1800,7 +1800,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1822,7 +1822,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1857,7 +1857,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1876,7 +1876,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1895,7 +1895,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2083,7 +2083,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2109,21 +2109,21 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
     course text(s). I haven't used it much for reading either, owing to the
     large amount of new notation one has to refresh every time you open the
@@ -2135,14 +2135,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
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 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -2151,14 +2151,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>408</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -2168,14 +2168,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -2230,28 +2230,28 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -2271,7 +2271,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -2287,7 +2287,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Bibliographical Note: This Dover edition, first published in 2006, is a
     slightly corrected unabridged republication of the revised (second) edition
     of the work originally published in 1984 by Elsevier Science Publishers
@@ -2301,7 +2301,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>416</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -2323,21 +2323,21 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -2373,14 +2373,14 @@ show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -2418,14 +2418,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>422</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -2453,7 +2453,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>424</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -2482,14 +2482,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -2512,7 +2512,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2577,14 +2577,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -2605,7 +2605,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -2637,14 +2637,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>432</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -2663,14 +2663,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>434</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>221</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -2683,7 +2683,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -2699,7 +2699,7 @@ show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 type="local"
@@ -2719,7 +2719,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>436</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -2728,7 +2728,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
     this list. Whatever else you might think about Quine, he was a terrific
     explainer, and this is the best introduction to the subject I know."
@@ -2742,7 +2742,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2810,7 +2810,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>440</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 toc="true"
@@ -2819,7 +2819,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -2835,14 +2835,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>442</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>443</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -2900,7 +2900,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2911,7 +2911,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2931,7 +2931,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>448</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2943,15 +2943,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>450</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        TODO: not a good selection
-    </fr:p><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Contributing to <fr:link
 type="external"
-href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+href="https://github.com/maxsnew/cubical-categorical-logic">Cubical Categorical Logic</fr:link>, "cubical agda formalizations of different approaches to categorical semantics of type theories"
         </fr:li>
         <fr:li><fr:link
 type="external"
-href="https://github.com/hejohns/.rc/tree/master/home/bin">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly
         </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
@@ -2959,7 +2958,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>451</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 type="external"
 href="https://mplse-reading-group.github.io/">https://mplse-reading-group.github.io/</fr:link></fr:li>
         <fr:li><fr:link
@@ -2971,7 +2970,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>452</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
 type="external"
 href="../posts.html">Jekyll Blog</fr:link></fr:mainmatter><fr:backmatter /></fr:tree>
     
@@ -2982,14 +2981,14 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>453</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3000,7 +2999,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>456</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
 type="external"
@@ -3011,7 +3010,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3030,7 +3029,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3044,7 +3043,7 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3074,7 +3073,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3090,7 +3089,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3105,7 +3104,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4260,7 +4259,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>461</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -4270,14 +4269,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>462</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -4332,7 +4331,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>463</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -4348,7 +4347,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>464</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -4370,14 +4369,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>465</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -4413,14 +4412,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>466</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -4458,7 +4457,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>467</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4471,14 +4470,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>469</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -4501,7 +4500,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4566,7 +4565,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4598,14 +4597,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>472</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -4626,7 +4625,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -4658,14 +4657,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>473</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -4684,7 +4683,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-178.xml b/_site/forest/unstable-178.xml
index 77ddc13..8f4ca6a 100644
--- a/_site/forest/unstable-178.xml
+++ b/_site/forest/unstable-178.xml
@@ -7,18 +7,18 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>478</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title><fr:link
 type="local"
 href="index.xml"
 addr="index"
-title="Johnson He">Johnson He</fr:link> › Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+title="Johnson He">Johnson He</fr:link> › Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -29,7 +29,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>456</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
 type="external"
@@ -40,7 +40,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -59,7 +59,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -73,7 +73,7 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -103,7 +103,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -119,7 +119,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -134,7 +134,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1289,7 +1289,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>461</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -1299,14 +1299,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>462</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -1361,7 +1361,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>463</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -1377,7 +1377,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>464</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -1399,14 +1399,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>465</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -1442,14 +1442,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>466</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -1487,7 +1487,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>467</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1500,14 +1500,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>469</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -1530,7 +1530,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1595,7 +1595,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1627,14 +1627,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>472</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -1655,7 +1655,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -1687,14 +1687,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>473</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -1713,7 +1713,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1746,7 +1746,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="true"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>479</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:meta
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>402</fr:anchor><fr:addr>index</fr:addr><fr:route>index.xml</fr:route><fr:title>Johnson He</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:meta
 name="author">false</fr:meta></fr:frontmatter><fr:mainmatter>
     
     <fr:tree
@@ -1756,7 +1756,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>219</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About Me</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -1798,14 +1798,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#173</fr:addr><fr:route>unstable-173.xml</fr:route><fr:title>Notes</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>223</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2960,7 +2960,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>270</fr:anchor><fr:addr>hejohns-0008</fr:addr><fr:route>hejohns-0008.xml</fr:route><fr:title>The core of adjoint functors</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2971,7 +2971,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>272</fr:anchor><fr:addr>hejohns-0006</fr:addr><fr:route>hejohns-0006.xml</fr:route><fr:title>The core of adjoint monotone functions</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -2988,7 +2988,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3015,7 +3015,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3041,7 +3041,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3072,7 +3072,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-0007</fr:addr><fr:route>hejohns-0007.xml</fr:route><fr:title>Precursor to The Core of Adjoint Functors</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3089,7 +3089,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3133,7 +3133,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3152,7 +3152,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3181,7 +3181,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3201,7 +3201,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3254,7 +3254,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3314,7 +3314,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3429,7 +3429,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3471,7 +3471,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3508,7 +3508,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>300</fr:anchor><fr:addr>hejohns-000E</fr:addr><fr:route>hejohns-000E.xml</fr:route><fr:title>Adjunction core</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3519,7 +3519,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3541,7 +3541,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3576,7 +3576,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3595,7 +3595,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3614,7 +3614,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3802,7 +3802,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -3828,21 +3828,21 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>403</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr>#174</fr:addr><fr:route>unstable-174.xml</fr:route><fr:title>References</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>404</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>shoenfield-1967</fr:addr><fr:route>shoenfield-1967.xml</fr:route><fr:title>Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>405</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>hinman-2005</fr:addr><fr:route>hinman-2005.xml</fr:route><fr:title>Fundamentals of Mathematical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The <fr:strong>tome</fr:strong>, sometimes used at Michigan, but it was never (one of) my
     course text(s). I haven't used it much for reading either, owing to the
     large amount of new notation one has to refresh every time you open the
@@ -3854,14 +3854,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>406</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>lambek-scott-1986</fr:addr><fr:route>lambek-scott-1986.xml</fr:route><fr:title>Introduction to higher order categorical logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>4</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>407</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>mclarty-1990a</fr:addr><fr:route>mclarty-1990a.xml</fr:route><fr:title>Book Review (1990) of John Bell's Toposes and Local Set Theories (1988)</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="mclarty-1990a.pdf">mclarty-1990a.pdf</fr:link></fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -3870,14 +3870,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>408</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>awodey-2010</fr:addr><fr:route>awodey-2010.xml</fr:route><fr:title>Category Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>409</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -3887,14 +3887,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>410</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -3949,28 +3949,28 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>411</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>maclane-moerdijk-1994</fr:addr><fr:route>maclane-moerdijk-1994.xml</fr:route><fr:title>Sheaves in Geometry and Logic: A First Introduction to Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>412</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>324</fr:anchor><fr:addr>johnstone-2002</fr:addr><fr:route>johnstone-2002.xml</fr:route><fr:title>Sketches of an Elephant: a topos theory compendium</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>413</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>325</fr:anchor><fr:addr>mclarty-1990b</fr:addr><fr:route>mclarty-1990b.xml</fr:route><fr:title>The Uses and Abuses of the History of Topos Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>326</fr:anchor><fr:addr>#170</fr:addr><fr:route>unstable-170.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>mclarty-1990b</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The view that toposes originated as generalized set theory is a figment of
         set theoretically educated common sense. This false history obstructs
         understanding of category theory and especially of categorical foundations
@@ -3990,7 +3990,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>414</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>327</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -4006,7 +4006,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>415</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>goldblatt-2006</fr:addr><fr:route>goldblatt-2006.xml</fr:route><fr:title>Topoi: The Categorial Analysis of Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Bibliographical Note: This Dover edition, first published in 2006, is a
     slightly corrected unabridged republication of the revised (second) edition
     of the work originally published in 1984 by Elsevier Science Publishers
@@ -4020,7 +4020,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>416</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>329</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -4042,21 +4042,21 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>417</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>mclarty-1992</fr:addr><fr:route>mclarty-1992.xml</fr:route><fr:title>Elementary Categories, Elementary Toposes</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter /><fr:backmatter /></fr:tree><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>418</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>331</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -4092,14 +4092,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>420</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>333</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -4137,14 +4137,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>422</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>335</fr:anchor><fr:addr>carette-kiselyov-shan-2007</fr:addr><fr:route>carette-kiselyov-shan-2007.xml</fr:route><fr:title>Finally Tagless, Partially Evaluated</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -4172,7 +4172,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>424</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
@@ -4201,14 +4201,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>338</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -4231,7 +4231,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4296,14 +4296,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>429</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -4324,7 +4324,7 @@ show-heading="true"
 show-metadata="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -4356,14 +4356,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>432</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>345</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -4382,14 +4382,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>434</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>347</fr:anchor><fr:addr>blass-1988</fr:addr><fr:route>blass-1988.xml</fr:route><fr:title>Topoi and Computation</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>221</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -4402,7 +4402,7 @@ show-heading="true"
 show-metadata="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
@@ -4418,7 +4418,7 @@ show-heading="true"
 show-metadata="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>435</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>350</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
 type="local"
@@ -4438,7 +4438,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>436</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>351</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="false"
@@ -4447,7 +4447,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>437</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>352</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
     this list. Whatever else you might think about Quine, he was a terrific
     explainer, and this is the best introduction to the subject I know."
@@ -4461,7 +4461,7 @@ show-heading="true"
 show-metadata="false"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>438</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4529,7 +4529,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>440</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>355</fr:anchor><fr:addr>street-2012</fr:addr><fr:route>street-2012.xml</fr:route><fr:title>The Core of Adjoint Functors</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     The first point:
 </fr:p><fr:tree
 toc="true"
@@ -4538,7 +4538,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>441</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>356</fr:anchor><fr:addr>#163</fr:addr><fr:route>unstable-163.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>street-2012</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         During my Topology lectures at Macquarie University in the 1970s, the
         students and I realized, in proving that a function f between posets was
         order preserving when there was a function u in the reverse direction such
@@ -4554,14 +4554,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>442</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>357</fr:anchor><fr:addr>maddy-2019</fr:addr><fr:route>maddy-2019.xml</fr:route><fr:title>What do we want a foundation to do?</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>443</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>358</fr:anchor><fr:addr>#171</fr:addr><fr:route>unstable-171.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>21</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>22</fr:day></fr:date><fr:authors /><fr:parent>maddy-2019</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Mainstream orthodoxy holds that set theory provides a foundation for
         contemporary pure mathematics. Critics of this view argue that category
         theory, or more recently univalent foundations, is better suited to this
@@ -4619,7 +4619,7 @@ show-heading="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>444</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>359</fr:anchor><fr:addr>dummett-91</fr:addr><fr:route>dummett-91.xml</fr:route><fr:title>The Logical Basis of Metaphysics</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4630,7 +4630,7 @@ show-heading="true"
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 expanded="true"
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-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4650,7 +4650,7 @@ show-heading="true"
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 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>448</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>jacobs-1999</fr:addr><fr:route>jacobs-1999.xml</fr:route><fr:title>Categorical Logic and Type Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4662,15 +4662,14 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>450</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        TODO: not a good selection
-    </fr:p><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>#175</fr:addr><fr:route>unstable-175.xml</fr:route><fr:title>Code</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li>
+            Contributing to <fr:link
 type="external"
-href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for EECS 490 grading, but also CSV, JSON, manipulation more broadly
+href="https://github.com/maxsnew/cubical-categorical-logic">Cubical Categorical Logic</fr:link>, "cubical agda formalizations of different approaches to categorical semantics of type theories"
         </fr:li>
         <fr:li><fr:link
 type="external"
-href="https://github.com/hejohns/.rc/tree/master/home/bin">misc cmdline utils</fr:link>. <fr:code>colorless</fr:code> is nice for long <fr:code>git status</fr:code> output (it's really color<fr:strong>-</fr:strong>less).
+href="https://github.com/eecs490/gradescope-utils">Gradescope-Utils</fr:link>, a suite of scripts for grading EECS 490 (programming languages), but also CSV, JSON, processing combinators more broadly
         </fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="true"
@@ -4678,7 +4677,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>451</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>366</fr:anchor><fr:addr>#176</fr:addr><fr:route>unstable-176.xml</fr:route><fr:title>Links</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:ul><fr:li><fr:link
 type="external"
 href="https://mplse-reading-group.github.io/">https://mplse-reading-group.github.io/</fr:link></fr:li>
         <fr:li><fr:link
@@ -4690,7 +4689,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>452</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>#177</fr:addr><fr:route>unstable-177.xml</fr:route><fr:title>Blog</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:link
 type="external"
 href="../posts.html">Jekyll Blog</fr:link></fr:mainmatter><fr:backmatter /></fr:tree>
     
@@ -4701,14 +4700,14 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>453</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>368</fr:anchor><fr:addr>#178</fr:addr><fr:route>unstable-178.xml</fr:route><fr:title>Inbox</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>7</fr:month><fr:day>2</fr:day></fr:date><fr:authors /><fr:parent>index</fr:parent></fr:frontmatter><fr:mainmatter><fr:tree
 toc="false"
 numbered="false"
 show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>454</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>369</fr:anchor><fr:addr>hejohns-000P</fr:addr><fr:route>hejohns-000P.xml</fr:route><fr:title>Logical Calculations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4719,7 +4718,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>456</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>hejohns-000Q</fr:addr><fr:route>hejohns-000Q.xml</fr:route><fr:title>Every (elementary) topos is regular</fr:title><fr:taxon>Lemma</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>16</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html">stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</fr:link></fr:p><fr:p><fr:link
 type="external"
@@ -4730,7 +4729,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>457</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>372</fr:anchor><fr:addr>hejohns-0005</fr:addr><fr:route>hejohns-0005.xml</fr:route><fr:title>Simulations</fr:title><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4749,7 +4748,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>374</fr:anchor><fr:addr>hejohns-0001</fr:addr><fr:route>hejohns-0001.xml</fr:route><fr:title>Simulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4763,7 +4762,7 @@ show-heading="true"
 show-metadata="false"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>376</fr:anchor><fr:addr>hejohns-0002</fr:addr><fr:route>hejohns-0002.xml</fr:route><fr:title>Bisimulation</fr:title><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4793,7 +4792,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>378</fr:anchor><fr:addr>hejohns-0003</fr:addr><fr:route>hejohns-0003.xml</fr:route><fr:title>Bisimilar Kripke frames are elementarily equivalent</fr:title><fr:taxon>Moral corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4809,7 +4808,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>380</fr:anchor><fr:addr>hejohns-0004</fr:addr><fr:route>hejohns-0004.xml</fr:route><fr:title>Preservation under simulations</fr:title><fr:taxon>Corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>4</fr:month><fr:day>6</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>6</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -4824,7 +4823,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>459</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>382</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -5979,7 +5978,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>461</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>384</fr:anchor><fr:addr>makkai-reyes-1977</fr:addr><fr:route>makkai-reyes-1977.xml</fr:route><fr:title>First Order Categorical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>2</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I've heard/been told this is <fr:em>the</fr:em> book for, well, the subject in the
     title, and Makkai completeness.
 </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
@@ -5989,14 +5988,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>462</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>lawvere-1970</fr:addr><fr:route>lawvere-1970.xml</fr:route><fr:title>Quantifiers and Sheaves</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>342</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#172</fr:addr><fr:route>unstable-172.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /><fr:parent>lawvere-1970</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The unity of opposites in the title is essentially that between logic
         and geometry, and there are compelling reasons for maintaining that
         geometry is the leading aspect. At the same lime, in the present joint
@@ -6051,7 +6050,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>463</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>386</fr:anchor><fr:addr>marquis-reyes-2012</fr:addr><fr:route>marquis-reyes-2012.xml</fr:route><fr:title>The history of categorical logic: 1963-1977</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     I have not read this. This might be a good source, aside from <fr:link
 type="local"
 href="ronnie-chen.xml"
@@ -6067,7 +6066,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>464</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>scherer-2017</fr:addr><fr:route>scherer-2017.xml</fr:route><fr:title>Deciding equivalence with sums and the empty type</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p><fr:link
 type="external"
 href="https://arxiv.org/pdf/1610.01213">https://arxiv.org/pdf/1610.01213</fr:link>.
     The nice introduction informed me of <fr:link
@@ -6089,14 +6088,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>465</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>388</fr:anchor><fr:addr>friedman-1975</fr:addr><fr:route>friedman-1975.xml</fr:route><fr:title>Equality Between Functionals</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
@@ -6132,14 +6131,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>466</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>389</fr:anchor><fr:addr>statman-1985</fr:addr><fr:route>statman-1985.xml</fr:route><fr:title>Equality Between Functionals Revisited</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>#166</fr:addr><fr:route>unstable-166.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>statman-1985</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         In this note we shall try to place Friedman's remarkable little paper
         (<fr:link
 type="local"
@@ -6177,7 +6176,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>467</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>390</fr:anchor><fr:addr>hejohns-000A</fr:addr><fr:route>hejohns-000A.xml</fr:route><fr:title>Fiberwise Structure</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -6190,14 +6189,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>469</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>392</fr:anchor><fr:addr>millies-zeilberger-2015</fr:addr><fr:route>millies-zeilberger-2015.xml</fr:route><fr:title>Functors are Type Refinement Systems</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>426</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>339</fr:anchor><fr:addr>#168</fr:addr><fr:route>unstable-168.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>millies-zeilberger-2015</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The standard reading of type theory through the lens of category theory
         is based on the idea of viewing a type system as a category of
         well-typed terms. We propose a basic revision of this reading: rather
@@ -6220,7 +6219,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>340</fr:anchor><fr:addr>#169</fr:addr><fr:route>unstable-169.xml</fr:route><fr:title>Sections 1-2</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -6285,7 +6284,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>470</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -6317,14 +6316,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>472</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>sterling-2022</fr:addr><fr:route>sterling-2022.xml</fr:route><fr:title>Naïve Logical Relations in Synthetic Tait Computability</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>430</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>343</fr:anchor><fr:addr>#164</fr:addr><fr:route>unstable-164.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Logical relations are the main tool for proving <fr:strong>positive</fr:strong>
         properties of logics, type theories, and programming languages:
         canonicity, decidability, conservativity, computational adequacy, and
@@ -6345,7 +6344,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>431</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>344</fr:anchor><fr:addr>#165</fr:addr><fr:route>unstable-165.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>sterling-2022</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         I can't quite follow the <fr:em>phase distinction</fr:em> and only have a very vague
         idea of what's going on starting at p. 9.
     </fr:p><fr:p>
@@ -6377,14 +6376,14 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>473</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>396</fr:anchor><fr:addr>blass-2003</fr:addr><fr:route>blass-2003.xml</fr:route><fr:title>Resource Consciousness in Classical Logic</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
@@ -6403,7 +6402,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>474</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>hejohns-0009</fr:addr><fr:route>hejohns-0009.xml</fr:route><fr:title>Fibrations v. functors</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>8</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-179.xml b/_site/forest/unstable-179.xml
index 65d260d..01d362f 100644
--- a/_site/forest/unstable-179.xml
+++ b/_site/forest/unstable-179.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>277</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>484</fr:anchor><fr:addr>#179</fr:addr><fr:route>unstable-179.xml</fr:route><fr:title><fr:link
 type="local"
 href="hejohns-000M.xml"
 addr="hejohns-000M"
@@ -177,7 +177,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>485</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-180.xml b/_site/forest/unstable-180.xml
index e85336a..c038dbb 100644
--- a/_site/forest/unstable-180.xml
+++ b/_site/forest/unstable-180.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>318</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>517</fr:anchor><fr:addr>#180</fr:addr><fr:route>unstable-180.xml</fr:route><fr:title><fr:link
 type="local"
 href="unstable-179.xml"
 addr="#179"
@@ -46,7 +46,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>319</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>518</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-181.xml b/_site/forest/unstable-181.xml
index 585cbed..541ef44 100644
--- a/_site/forest/unstable-181.xml
+++ b/_site/forest/unstable-181.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>511</fr:anchor><fr:addr>#181</fr:addr><fr:route>unstable-181.xml</fr:route><fr:title><fr:link
 type="local"
 href="unstable-179.xml"
 addr="#179"
@@ -118,7 +118,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>313</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>512</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-182.xml b/_site/forest/unstable-182.xml
index 1ee556f..f9c6a23 100644
--- a/_site/forest/unstable-182.xml
+++ b/_site/forest/unstable-182.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>501</fr:anchor><fr:addr>#182</fr:addr><fr:route>unstable-182.xml</fr:route><fr:title><fr:link
 type="local"
 href="unstable-179.xml"
 addr="#179"
@@ -44,7 +44,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>295</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>502</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-183.xml b/_site/forest/unstable-183.xml
index eb6114a..9bdc462 100644
--- a/_site/forest/unstable-183.xml
+++ b/_site/forest/unstable-183.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>279</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>486</fr:anchor><fr:addr>#183</fr:addr><fr:route>unstable-183.xml</fr:route><fr:title><fr:link
 type="local"
 href="hejohns-000M.xml"
 addr="hejohns-000M"
@@ -253,7 +253,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>280</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>487</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-184.xml b/_site/forest/unstable-184.xml
index cf69220..fa6385a 100644
--- a/_site/forest/unstable-184.xml
+++ b/_site/forest/unstable-184.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>320</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>519</fr:anchor><fr:addr>#184</fr:addr><fr:route>unstable-184.xml</fr:route><fr:title><fr:link
 type="local"
 href="unstable-183.xml"
 addr="#183"
@@ -94,7 +94,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>321</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>520</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-185.xml b/_site/forest/unstable-185.xml
index 23eb1b0..f8058fe 100644
--- a/_site/forest/unstable-185.xml
+++ b/_site/forest/unstable-185.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>314</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>513</fr:anchor><fr:addr>#185</fr:addr><fr:route>unstable-185.xml</fr:route><fr:title><fr:link
 type="local"
 href="unstable-183.xml"
 addr="#183"
@@ -120,7 +120,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>315</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>514</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-186.xml b/_site/forest/unstable-186.xml
index 997894e..c1d304e 100644
--- a/_site/forest/unstable-186.xml
+++ b/_site/forest/unstable-186.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>503</fr:anchor><fr:addr>#186</fr:addr><fr:route>unstable-186.xml</fr:route><fr:title><fr:link
 type="local"
 href="unstable-183.xml"
 addr="#183"
@@ -63,7 +63,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>297</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>504</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-187.xml b/_site/forest/unstable-187.xml
index 3a24f4c..22f31f2 100644
--- a/_site/forest/unstable-187.xml
+++ b/_site/forest/unstable-187.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>281</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>488</fr:anchor><fr:addr>#187</fr:addr><fr:route>unstable-187.xml</fr:route><fr:title><fr:link
 type="local"
 href="hejohns-000M.xml"
 addr="hejohns-000M"
@@ -149,7 +149,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>489</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-188.xml b/_site/forest/unstable-188.xml
index 9c2b9d4..e353cff 100644
--- a/_site/forest/unstable-188.xml
+++ b/_site/forest/unstable-188.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>322</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>521</fr:anchor><fr:addr>#188</fr:addr><fr:route>unstable-188.xml</fr:route><fr:title><fr:link
 type="local"
 href="unstable-187.xml"
 addr="#187"
@@ -37,7 +37,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>323</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>522</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-189.xml b/_site/forest/unstable-189.xml
index 54effa6..40a4c78 100644
--- a/_site/forest/unstable-189.xml
+++ b/_site/forest/unstable-189.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>316</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>515</fr:anchor><fr:addr>#189</fr:addr><fr:route>unstable-189.xml</fr:route><fr:title><fr:link
 type="local"
 href="unstable-187.xml"
 addr="#187"
@@ -88,7 +88,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>317</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>516</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-190.xml b/_site/forest/unstable-190.xml
index 7ec8005..f627dd6 100644
--- a/_site/forest/unstable-190.xml
+++ b/_site/forest/unstable-190.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>505</fr:anchor><fr:addr>#190</fr:addr><fr:route>unstable-190.xml</fr:route><fr:title><fr:link
 type="local"
 href="unstable-187.xml"
 addr="#187"
@@ -64,7 +64,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>299</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>506</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-191.xml b/_site/forest/unstable-191.xml
index 6bda65a..5b62c29 100644
--- a/_site/forest/unstable-191.xml
+++ b/_site/forest/unstable-191.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>509</fr:anchor><fr:addr>#191</fr:addr><fr:route>unstable-191.xml</fr:route><fr:title><fr:link
 type="local"
 href="hejohns-000M.xml"
 addr="hejohns-000M"
@@ -57,7 +57,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>311</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>510</fr:anchor><fr:addr>hejohns-000M</fr:addr><fr:route>hejohns-000M.xml</fr:route><fr:title>Finite-type intuitionistic predicate logic</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>3</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     There is some (a posteriori insignificant) variation in the presentation of
     the syntax, thus following through with some input from <fr:link
 type="local"
diff --git a/_site/forest/unstable-201.xml b/_site/forest/unstable-201.xml
index be7ad8f..b23042c 100644
--- a/_site/forest/unstable-201.xml
+++ b/_site/forest/unstable-201.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>531</fr:anchor><fr:addr>#201</fr:addr><fr:route>unstable-201.xml</fr:route><fr:title><fr:link
 type="local"
 href="hejohns-000C.xml"
 addr="hejohns-000C"
@@ -80,7 +80,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>334</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>533</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-202.xml b/_site/forest/unstable-202.xml
index a250e3c..2946de8 100644
--- a/_site/forest/unstable-202.xml
+++ b/_site/forest/unstable-202.xml
@@ -7,7 +7,7 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>328</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>527</fr:anchor><fr:addr>#202</fr:addr><fr:route>unstable-202.xml</fr:route><fr:title><fr:link
 type="local"
 href="hejohns-000C.xml"
 addr="hejohns-000C"
@@ -1100,7 +1100,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>330</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>529</fr:anchor><fr:addr>hejohns-000C</fr:addr><fr:route>hejohns-000C.xml</fr:route><fr:title>Elementary questions, elementary toposes</fr:title><fr:taxon>Exposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>31</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>15</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-203.xml b/_site/forest/unstable-203.xml
index fd0222f..ae740c0 100644
--- a/_site/forest/unstable-203.xml
+++ b/_site/forest/unstable-203.xml
@@ -65,7 +65,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>371</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>282</fr:anchor><fr:addr>#203</fr:addr><fr:route>unstable-203.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -109,7 +109,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>373</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>284</fr:anchor><fr:addr>sub-0007</fr:addr><fr:route>sub-0007.xml</fr:route><fr:title>The short proof (sketch)</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -128,7 +128,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>375</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>286</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -157,7 +157,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>377</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>288</fr:anchor><fr:addr>subsubb-0007</fr:addr><fr:route>subsubb-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -177,7 +177,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>379</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>290</fr:anchor><fr:addr>hejohns-000D</fr:addr><fr:route>hejohns-000D.xml</fr:route><fr:title>The verbose proof</fr:title><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -230,7 +230,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>292</fr:anchor><fr:addr>subb-000D</fr:addr><fr:route>subb-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -290,7 +290,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>383</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>294</fr:anchor><fr:addr>sub-000D</fr:addr><fr:route>sub-000D.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -405,7 +405,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>385</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>296</fr:anchor><fr:addr>subsub-0007</fr:addr><fr:route>subsub-0007.xml</fr:route><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -447,7 +447,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>387</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>298</fr:anchor><fr:addr>subbb-000D</fr:addr><fr:route>subbb-000D.xml</fr:route><fr:taxon>Moral</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-204.xml b/_site/forest/unstable-204.xml
index 66159fe..2bf7209 100644
--- a/_site/forest/unstable-204.xml
+++ b/_site/forest/unstable-204.xml
@@ -58,7 +58,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>363</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>274</fr:anchor><fr:addr>#204</fr:addr><fr:route>unstable-204.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -85,7 +85,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>365</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>276</fr:anchor><fr:addr>sub-0006</fr:addr><fr:route>sub-0006.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -111,7 +111,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>367</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>278</fr:anchor><fr:addr>subb-0006</fr:addr><fr:route>subb-0006.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>7</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-205.xml b/_site/forest/unstable-205.xml
index 557d2a8..cf11d01 100644
--- a/_site/forest/unstable-205.xml
+++ b/_site/forest/unstable-205.xml
@@ -7,11 +7,11 @@ show-heading="true"
 show-metadata="true"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>496</fr:anchor><fr:addr>#205</fr:addr><fr:route>unstable-205.xml</fr:route><fr:title><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>421</fr:anchor><fr:addr>#205</fr:addr><fr:route>unstable-205.xml</fr:route><fr:title><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
-title="Johnson He">Johnson He</fr:link> › Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+title="Johnson He">Johnson He</fr:link> › Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -33,7 +33,7 @@ show-heading="true"
 show-metadata="true"
 expanded="false"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>498</fr:anchor><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:taxon>Person</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>hejohns</fr:addr><fr:route>hejohns.xml</fr:route><fr:title>Johnson He</fr:title><fr:taxon>Person</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -46,7 +46,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>492</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>425</fr:anchor><fr:addr>hejohns-000O</fr:addr><fr:route>hejohns-000O.xml</fr:route><fr:title>Short About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -81,7 +81,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>494</fr:anchor><fr:addr>#205</fr:addr><fr:route>unstable-205.xml</fr:route><fr:title>Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>10</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>427</fr:anchor><fr:addr>#205</fr:addr><fr:route>unstable-205.xml</fr:route><fr:title>Extended About</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>5</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-206.xml b/_site/forest/unstable-206.xml
index 14bdcb2..419655c 100644
--- a/_site/forest/unstable-206.xml
+++ b/_site/forest/unstable-206.xml
@@ -53,7 +53,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>419</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>332</fr:anchor><fr:addr>#206</fr:addr><fr:route>unstable-206.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>friedman-1975</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         The λ-calculus, in both its typed and untyped forms, has primarily been
         regarded as an attempt to formalize the concept of <fr:em>rule</fr:em> or
         <fr:em>process</fr:em>, and ultimately to provide a new foundation for mathematics.
diff --git a/_site/forest/unstable-207.xml b/_site/forest/unstable-207.xml
index 842d0f2..4c92187 100644
--- a/_site/forest/unstable-207.xml
+++ b/_site/forest/unstable-207.xml
@@ -49,7 +49,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>446</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>361</fr:anchor><fr:addr>#207</fr:addr><fr:route>unstable-207.xml</fr:route><fr:title>Stability</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>12</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>13</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-208.xml b/_site/forest/unstable-208.xml
index 666888d..ab4c911 100644
--- a/_site/forest/unstable-208.xml
+++ b/_site/forest/unstable-208.xml
@@ -49,7 +49,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -77,7 +77,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>424</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
diff --git a/_site/forest/unstable-209.xml b/_site/forest/unstable-209.xml
index b2f0ba5..f78b1e8 100644
--- a/_site/forest/unstable-209.xml
+++ b/_site/forest/unstable-209.xml
@@ -50,7 +50,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>423</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>336</fr:anchor><fr:addr>#208</fr:addr><fr:route>unstable-208.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         We have built the first family of tagless interpretations for a
         higher-order typed object language in a typed metalanguage (Haskell or
         ML) that require no dependent types, generalized algebraic data types,
@@ -78,7 +78,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>424</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>337</fr:anchor><fr:addr>#209</fr:addr><fr:route>unstable-209.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>24</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>25</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>carette-kiselyov-shan-2007</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the APLAS version (conference paper). There is a roughly twice
         as long journal version.
     </fr:p><fr:p>
diff --git a/_site/forest/unstable-210.xml b/_site/forest/unstable-210.xml
index e50e803..4fb436d 100644
--- a/_site/forest/unstable-210.xml
+++ b/_site/forest/unstable-210.xml
@@ -35,7 +35,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>433</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>346</fr:anchor><fr:addr>#210</fr:addr><fr:route>unstable-210.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-2003</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Using Herbrand's Theorem, we define simple Herbrand validity, a sort of
         resource consciousness that makes sense in classical predicate logic.
         We characterize the propositional formulas all of whose first-order
diff --git a/_site/forest/unstable-211.xml b/_site/forest/unstable-211.xml
index ad92c51..345c42f 100644
--- a/_site/forest/unstable-211.xml
+++ b/_site/forest/unstable-211.xml
@@ -34,7 +34,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>221</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -47,7 +47,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
diff --git a/_site/forest/unstable-212.xml b/_site/forest/unstable-212.xml
index d0bb7c0..f431e51 100644
--- a/_site/forest/unstable-212.xml
+++ b/_site/forest/unstable-212.xml
@@ -35,7 +35,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>221</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>348</fr:anchor><fr:addr>#211</fr:addr><fr:route>unstable-211.xml</fr:route><fr:taxon>Abstract</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         This is the written version of a talk given at the C.I.R.M. Workshop on
         Logic in Computer Science at Marseille, France in June, 1988. Its purpose
         is to argue that there is a close connection between the theory of
@@ -48,7 +48,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>222</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>349</fr:anchor><fr:addr>#212</fr:addr><fr:route>unstable-212.xml</fr:route><fr:taxon>Notes</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors /><fr:parent>blass-1988</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Prof. Blass essentially described this background over the course of
         the logic seminar last semester (as comments at various meetings,
         culminating with a full talk), but this is a nice written account of
diff --git a/_site/forest/unstable-213.xml b/_site/forest/unstable-213.xml
index f7c6ebe..f42f6d1 100644
--- a/_site/forest/unstable-213.xml
+++ b/_site/forest/unstable-213.xml
@@ -47,7 +47,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -69,7 +69,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -104,7 +104,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -123,7 +123,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -142,7 +142,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -330,7 +330,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-214.xml b/_site/forest/unstable-214.xml
index ce4d165..96addcd 100644
--- a/_site/forest/unstable-214.xml
+++ b/_site/forest/unstable-214.xml
@@ -42,7 +42,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -61,7 +61,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -98,7 +98,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -120,7 +120,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -155,7 +155,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -174,7 +174,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -193,7 +193,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -381,7 +381,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/unstable-215.xml b/_site/forest/unstable-215.xml
index d77ca88..83d6614 100644
--- a/_site/forest/unstable-215.xml
+++ b/_site/forest/unstable-215.xml
@@ -213,7 +213,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>391</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>302</fr:anchor><fr:addr>#213</fr:addr><fr:route>unstable-213.xml</fr:route><fr:taxon>Definition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -235,7 +235,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>393</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>304</fr:anchor><fr:addr>#214</fr:addr><fr:route>unstable-214.xml</fr:route><fr:taxon>Proposition</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -270,7 +270,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>395</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>306</fr:anchor><fr:addr>subb-000E</fr:addr><fr:route>subb-000E.xml</fr:route><fr:title>I</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -289,7 +289,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>397</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>308</fr:anchor><fr:addr>subbb-000E</fr:addr><fr:route>subbb-000E.xml</fr:route><fr:title>II</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -308,7 +308,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>399</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>310</fr:anchor><fr:addr>#215</fr:addr><fr:route>unstable-215.xml</fr:route><fr:taxon>Proof</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
@@ -496,7 +496,7 @@ show-heading="true"
 show-metadata="false"
 expanded="true"
 root="false"
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>401</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>312</fr:anchor><fr:addr>sub-000E</fr:addr><fr:route>sub-000E.xml</fr:route><fr:taxon>Proof corollary</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>11</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
 href="hejohns.xml"
 addr="hejohns"
diff --git a/_site/forest/van-heijenoort-1967.xml b/_site/forest/van-heijenoort-1967.xml
index 4bca898..23d026b 100644
--- a/_site/forest/van-heijenoort-1967.xml
+++ b/_site/forest/van-heijenoort-1967.xml
@@ -1,30 +1,39 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>561</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>749</fr:anchor><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
+numbered="false"
+show-heading="true"
+show-metadata="false"
+expanded="true"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>560</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>750</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
-href="quine-1960.xml"
 type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
+href="quine-1960.xml"
+addr="quine-1960"
+title="Variables Explained Away">Variables Explained Away</fr:link>.
 </fr:p><fr:p>
     English translation in <fr:link
-href="van-heijenoort-1967.xml"
 type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="van-heijenoort-1967.xml"
+addr="van-heijenoort-1967"
+title="From Frege To Gödel">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/forest/von-neumann-1925.xml b/_site/forest/von-neumann-1925.xml
index 2a8bedb..b4e6d31 100644
--- a/_site/forest/von-neumann-1925.xml
+++ b/_site/forest/von-neumann-1925.xml
@@ -1,118 +1,49 @@
 <?xml version="1.0" encoding="UTF-8"?>
 <?xml-stylesheet type="text/xsl" href="default.xsl"?>
 <fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="true"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>559</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>745</fr:anchor><fr:addr>von-neumann-1925</fr:addr><fr:route>von-neumann-1925.xml</fr:route><fr:title>An Axiomatization of Set Theory</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     One of the early NBG papers. Maybe "the" NBG paper.
     My notes on this are currently in <fr:link
-href="quine-1960.xml"
 type="local"
-addr="quine-1960">Variables Explained Away</fr:link>.
+href="quine-1960.xml"
+addr="quine-1960"
+title="Variables Explained Away">Variables Explained Away</fr:link>.
 </fr:p><fr:p>
     English translation in <fr:link
-href="van-heijenoort-1967.xml"
 type="local"
-addr="van-heijenoort-1967">From Frege To Gödel</fr:link>.
-</fr:p></fr:mainmatter><fr:backmatter><fr:contributions></fr:contributions><fr:context></fr:context><fr:related></fr:related><fr:backlinks><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
+href="van-heijenoort-1967.xml"
+addr="van-heijenoort-1967"
+title="From Frege To Gödel">From Frege To Gödel</fr:link>.
+</fr:p></fr:mainmatter><fr:backmatter><fr:tree
+toc="false"
 numbered="false"
 show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>556</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
-    this list. Whatever else you might think about Quine, he was a terrific
-    explainer, and this is the best introduction to the subject I know."
-</fr:p><fr:p>
-    (Yes, that 50 should be a 60(b).)
-    Here is the followup I wrote:
-</fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="true"
-show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
-type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
-        Thanks for sharing this nice little note, which sketches a combinatory
-        (classical) first-order logic.
-    </fr:p><fr:p>
-        I was surprised by the mention of von Neumann 1925 in this snippet:
-    </fr:p><fr:p>
-        "In a way our operators are reminiscent also of the axioms of class
-        existence in Neumann, J. von, Eine Axiomatisierung der Mengenlehre,
-        Journal für reine und ungewandte Mathematik 154: 219-240, 1925, despite
-        dissimilarity of purpose." (Footnote 1, p. 346)
-    </fr:p><fr:p>
-        From what I gather, the reminiscence with von Neumann 1925 (I assume
-        Quine was principally referring to the "III. Logical Construction
-        Axioms") is that the purposes (of Quine's predicate combinators and von
-        Neumann's axioms of existence of II-objects) are actually quite
-        similar.
-    </fr:p><fr:p>
-        It seems that the system of von Neumann 1925 isn't couched in any
-        particular formal system (although it's maybe implicitly first-order,
-        and apparently translated into such by Skolem 1938), thus von Neumann
-        wants to recover from any definite property ("logical condition") on
-        I-objects, the (existence of a) class (II-object) of those I-objects.
-        Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
-    </fr:p><fr:p>
-        (More generally, the "reduction theorem" in von Neumann 1925 says that
-        every expression of I-objects (definable class-function) is indeed a
-        II-object (class-function). This restricts to logical conditions and
-        classes.)
-    </fr:p><fr:p>
-        To do this economically, the existence of a few basic
-        combinator-classes are stipulated (=, ∀, ∃!), and more complex logical
-        conditions are put in "normal form" by eliminating variables with the
-        combinator-classes.  The 'uneconomic' way, if von Neumann 1925 were
-        working in an explicit logic, would be to just stipulate a schema of
-        class comprehension (also mentioned in the Wikipedia article), as
-        included in the back of Chang and Keisler.
-    </fr:p><fr:p>
-        Actually, while there are explicit class existence axioms for absorbing
-        =, ∀, ∃!, I don't see any for things like ∧ or ¬, and don't see how
-        they'd follow with the other axioms, but I suppose they must. The
-        Wikipedia article has a write-up starting from more user-friendly
-        axioms.
-    </fr:p><fr:p>
-        In any case, I hadn't realized that the historic move to make precise
-        the notion of 'definite property' as 'first-order definable' was
-        nontrivial, and found this connection with Quine's note interesting,
-    </fr:p><fr:p>
-        Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
-    <fr:li><fr:link
-href="von-neumann-1925.xml"
-type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:title>Backlinks</fr:title><fr:authors /></fr:frontmatter><fr:mainmatter><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>557</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>746</fr:anchor><fr:addr>hejohns-000B</fr:addr><fr:route>hejohns-000B.xml</fr:route><fr:title>Logical Frameworks as a meta-language</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>27</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>6</fr:month><fr:day>1</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors></fr:frontmatter><fr:mainmatter><fr:p>
     While reading <fr:link
-href="von-neumann-1925.xml"
 type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link>, I just had the completely trivial-- and
     "that's… the point"-- "realization" that first order logic is the
     meta-language of axiomatic mathematics, just as (it seems) logical
     frameworks are the meta-languages of (typically λ-calculus based)
@@ -121,27 +52,21 @@ addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link>, I just had th
 </fr:p><fr:p>
     That is, (classical) FoL is often the setting in which axiomatic
     mathematical theories are formalized. Classic (no pun intended) examples
-    are <fr:tex>( \mathcal {L}_{group} =  \{ e, ⋅, -⁻¹ \} , Th( \textsf {groups}))</fr:tex>, <fr:tex>( \mathcal {L}_{DLO} =  \{ ≤ \} , Th( \textsf {Dense Linear Orders}))</fr:tex>, <fr:tex>( \mathcal {L}_{sets} =  \{ ∈ \} , ZFC)</fr:tex>.
+    are <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{group} = \{e, ⋅, -⁻¹\}, Th(\textsf {groups}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{DLO} = \{≤\}, Th(\textsf {Dense Linear Orders}))]]></fr:tex>, <fr:tex
+display="inline"><![CDATA[(\mathcal {L}_{sets} = \{∈\}, ZFC)]]></fr:tex>.
     (To be ultra-pedantic is a fruitless way, I've paired the language and theory.)
 </fr:p><fr:p>
     "definite property"
-</fr:p></fr:mainmatter></fr:tree></fr:backlinks><fr:references><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
+</fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:tree
 toc="true"
 numbered="false"
 show-heading="true"
 show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>554</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>van-heijenoort-1967</fr:addr><fr:route>van-heijenoort-1967.xml</fr:route><fr:title>From Frege To Gödel</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
-    A+. I should get a second copy to gift whenever needed.
-</fr:p></fr:mainmatter></fr:tree><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
-toc="true"
-numbered="false"
-show-heading="true"
-show-metadata="true"
-expanded="true"
-root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:taxon>Reference</fr:taxon><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date></fr:frontmatter><fr:mainmatter><fr:p>
+expanded="false"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>748</fr:anchor><fr:addr>quine-1960</fr:addr><fr:route>quine-1960.xml</fr:route><fr:title>Variables Explained Away</fr:title><fr:taxon>Reference</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors /></fr:frontmatter><fr:mainmatter><fr:p>
     "Separately I sent a copy of Quine's 1950 paper on combinatory logic to
     this list. Whatever else you might think about Quine, he was a terrific
     explainer, and this is the best introduction to the subject I know."
@@ -149,16 +74,17 @@ root="false"><fr:frontmatter><fr:anchor>555</fr:anchor><fr:taxon>Reference</fr:t
     (Yes, that 50 should be a 60(b).)
     Here is the followup I wrote:
 </fr:p><fr:tree
-xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"
 toc="true"
 numbered="true"
 show-heading="true"
 show-metadata="false"
 expanded="true"
-root="false"><fr:frontmatter><fr:anchor>381</fr:anchor><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
-href="hejohns.xml"
+root="false"
+xmlns:fr="http://www.jonmsterling.com/jms-005P.xml"><fr:frontmatter><fr:anchor>353</fr:anchor><fr:addr>#167</fr:addr><fr:route>unstable-167.xml</fr:route><fr:taxon>Email</fr:taxon><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>28</fr:day></fr:date><fr:date><fr:year>2024</fr:year><fr:month>5</fr:month><fr:day>29</fr:day></fr:date><fr:authors><fr:author><fr:link
 type="local"
-addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
+href="hejohns.xml"
+addr="hejohns"
+title="Johnson He">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-1960</fr:parent></fr:frontmatter><fr:mainmatter><fr:p>
         Thanks for sharing this nice little note, which sketches a combinatory
         (classical) first-order logic.
     </fr:p><fr:p>
@@ -181,8 +107,8 @@ addr="hejohns">Johnson He</fr:link></fr:author></fr:authors><fr:parent>quine-196
         wants to recover from any definite property ("logical condition") on
         I-objects, the (existence of a) class (II-object) of those I-objects.
         Wikipedia calls this the <fr:link
-href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem"
-type="external">class existence theorem</fr:link>.
+type="external"
+href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory#Class_existence_theorem">class existence theorem</fr:link>.
     </fr:p><fr:p>
         (More generally, the "reduction theorem" in von Neumann 1925 says that
         every expression of I-objects (definable class-function) is indeed a
@@ -208,10 +134,11 @@ type="external">class existence theorem</fr:link>.
         nontrivial, and found this connection with Quine's note interesting,
     </fr:p><fr:p>
         Johnson He
-    </fr:p></fr:mainmatter></fr:tree><fr:ul><fr:li><fr:link
-href="quine-1960.pdf"
-type="external">quine-1960.pdf</fr:link></fr:li>
+    </fr:p></fr:mainmatter><fr:backmatter /></fr:tree><fr:ul><fr:li><fr:link
+type="external"
+href="quine-1960.pdf">quine-1960.pdf</fr:link></fr:li>
     <fr:li><fr:link
-href="von-neumann-1925.xml"
 type="local"
-addr="von-neumann-1925">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter></fr:tree></fr:references></fr:backmatter></fr:tree>
\ No newline at end of file
+href="von-neumann-1925.xml"
+addr="von-neumann-1925"
+title="An Axiomatization of Set Theory">An Axiomatization of Set Theory</fr:link></fr:li></fr:ul></fr:mainmatter><fr:backmatter /></fr:tree></fr:mainmatter><fr:backmatter /></fr:tree></fr:backmatter></fr:tree>
\ No newline at end of file
diff --git a/_site/notes/1-25.html b/_site/notes/1-25.html
index 4537497..e54668a 100644
--- a/_site/notes/1-25.html
+++ b/_site/notes/1-25.html
@@ -307,7 +307,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/notes/2-08.html b/_site/notes/2-08.html
index c7fb19f..746d80b 100644
--- a/_site/notes/2-08.html
+++ b/_site/notes/2-08.html
@@ -291,7 +291,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/posts.html b/_site/posts.html
index 9ce2682..97bbce0 100644
--- a/_site/posts.html
+++ b/_site/posts.html
@@ -439,7 +439,7 @@ <h3>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/reviews.html b/_site/reviews.html
index b7bf52a..575a17d 100644
--- a/_site/reviews.html
+++ b/_site/reviews.html
@@ -69,8 +69,8 @@ <h1 class="post-title">reviews</h1>
 
 <p>Shoutout to kaobook.</p>
 
-<style> .pdf-embed-wrap-3b746764-0fa8-4b31-88cc-5395877252d9 { display: flex; flex-direction: column; width: 100%; height: 650px; } .pdf-embed-container-3b746764-0fa8-4b31-88cc-5395877252d9 { height: 100%; } .pdf-link-3b746764-0fa8-4b31-88cc-5395877252d9 { background-color: white; text-align: center; border-style: solid; } .pdf-embed-container-3b746764-0fa8-4b31-88cc-5395877252d9 iframe { width: 100%; height: 100%; } </style>
-<div class="pdf-embed-wrap-3b746764-0fa8-4b31-88cc-5395877252d9"> <div class="pdf-link-3b746764-0fa8-4b31-88cc-5395877252d9"> <a href="/files/sample.pdf" target="_blank">View PDF</a> </div> <div class="pdf-embed-container-3b746764-0fa8-4b31-88cc-5395877252d9"> <iframe src="/files/sample.pdf" frameborder="0" allowfullscreen=""></iframe> </div> </div>
+<style> .pdf-embed-wrap-9649ea4e-c686-43a4-949b-1cf1de674a4a { display: flex; flex-direction: column; width: 100%; height: 650px; } .pdf-embed-container-9649ea4e-c686-43a4-949b-1cf1de674a4a { height: 100%; } .pdf-link-9649ea4e-c686-43a4-949b-1cf1de674a4a { background-color: white; text-align: center; border-style: solid; } .pdf-embed-container-9649ea4e-c686-43a4-949b-1cf1de674a4a iframe { width: 100%; height: 100%; } </style>
+<div class="pdf-embed-wrap-9649ea4e-c686-43a4-949b-1cf1de674a4a"> <div class="pdf-link-9649ea4e-c686-43a4-949b-1cf1de674a4a"> <a href="/files/sample.pdf" target="_blank">View PDF</a> </div> <div class="pdf-embed-container-9649ea4e-c686-43a4-949b-1cf1de674a4a"> <iframe src="/files/sample.pdf" frameborder="0" allowfullscreen=""></iframe> </div> </div>
 
   </div>
 
@@ -99,7 +99,7 @@ <h1 class="post-title">reviews</h1>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction.html b/_site/site-construction.html
index d9b9bd7..c831a7a 100644
--- a/_site/site-construction.html
+++ b/_site/site-construction.html
@@ -124,7 +124,7 @@ <h3 id="specs">specs</h3>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/04/05/katex-test.html b/_site/site-construction/2023/04/05/katex-test.html
index 1b88717..8a5885d 100644
--- a/_site/site-construction/2023/04/05/katex-test.html
+++ b/_site/site-construction/2023/04/05/katex-test.html
@@ -140,7 +140,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/04/05/last-modified-time-test.html b/_site/site-construction/2023/04/05/last-modified-time-test.html
index dc557ea..fbecb5a 100644
--- a/_site/site-construction/2023/04/05/last-modified-time-test.html
+++ b/_site/site-construction/2023/04/05/last-modified-time-test.html
@@ -128,7 +128,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/04/05/page-sorting.html b/_site/site-construction/2023/04/05/page-sorting.html
index 6e967ec..845d4f9 100644
--- a/_site/site-construction/2023/04/05/page-sorting.html
+++ b/_site/site-construction/2023/04/05/page-sorting.html
@@ -125,7 +125,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/04/05/welcome-to-jekyll.html b/_site/site-construction/2023/04/05/welcome-to-jekyll.html
index acf6e17..d7f60fb 100644
--- a/_site/site-construction/2023/04/05/welcome-to-jekyll.html
+++ b/_site/site-construction/2023/04/05/welcome-to-jekyll.html
@@ -122,7 +122,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/04/07/underscore-plugins.html b/_site/site-construction/2023/04/07/underscore-plugins.html
index dbf3605..aeb3f36 100644
--- a/_site/site-construction/2023/04/07/underscore-plugins.html
+++ b/_site/site-construction/2023/04/07/underscore-plugins.html
@@ -138,7 +138,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/05/14/darcs.html b/_site/site-construction/2023/05/14/darcs.html
index 890f40e..072a023 100644
--- a/_site/site-construction/2023/05/14/darcs.html
+++ b/_site/site-construction/2023/05/14/darcs.html
@@ -117,7 +117,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/site-construction/2023/11/06/live-reload.html b/_site/site-construction/2023/11/06/live-reload.html
index fb8243b..4c6af93 100644
--- a/_site/site-construction/2023/11/06/live-reload.html
+++ b/_site/site-construction/2023/11/06/live-reload.html
@@ -118,7 +118,7 @@ <h2> related </h2>
             <li>(the answer is yes, you should.)</li>
         </ul>
         <!-- add last-modified-time -->
-        <p>site version: 2024-06-29</p>
+        <p>site version: 2024-07-02</p>
       </div>
       <div class="footer-col">
         <p>a pure waste of space</p>
diff --git a/_site/sitemap.xml b/_site/sitemap.xml
index 828e149..d58b019 100644
--- a/_site/sitemap.xml
+++ b/_site/sitemap.xml
@@ -848,87 +848,87 @@
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/blass-1988.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/blass-2003.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/carette-kiselyov-shan-2007.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/friedman-1975.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/hoffman-1999.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/lambek-1989.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/lawvere-1970.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/maddy-2019.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/marquis-reyes-2012.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/mclarty-1990a.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/mclarty-1990b.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/millies-zeilberger-2015.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/pfenning-elliott-1988.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/quine-1960.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular.html</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular_files/a.html</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/stack-overflow/elementary-proof-that-in-a-topos-every-epimorphism-is-regular_files/select.html</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/stack-overflow/every-topos-is-a-regular-category.html</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/statman-1985.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/sterling-2022.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/forest/wadler-1989-ocr.pdf</loc>
-<lastmod>2024-06-29T04:12:54-04:00</lastmod>
+<lastmod>2024-07-02T03:34:01-04:00</lastmod>
 </url>
 <url>
 <loc>https://hejohns.github.io/google1be2851b772a6a30.html</loc>